Evolution Equations of Random Walk and Poisson Processes

Gauge Institute Journal,H. Vic DannonEvolution Equations ofRandom Walk, and PoissonProcessesH. Vic Dannonvic0@comcast.netMarch, 2013Abstract A Random Differential Equation is the evolutionequation of a Random Process X(,)ζ t , driven by B(,) ζ t ,where B(,)ζ t is Random Walk, or by P (,) ζ t , where P(,ζ t)is aPoisson Process.For instance, the Langevin equation for the linear oscillatoris a first order evolution equation for the linear oscillatorprocess, driven by B(,) ζ t . To date, due to the confusionsurrounding Random Differential Equations, only thatequation was integrated and given a Random Processsolution.Integrating the probability-wave equation associated withthe random processX(,)ζ tis equivalent to integrating theRandom Differential Equation for the time-evolution of the1

Gauge Institute Journal,H. Vic DannonProcess X(,)ζ t .But the evolution equation solution is superior to the setup,and solution of the probability-wave equation:While the probability-wave equation setup is instructive,[Dan6], it involves Conditional Probabilities that for higherorder equations are beyond comprehension, reminding ofthe Monty Hall Problem [Rosenhouse].Consequently, it will be difficult to detect an error in thesetup of the probability-wave equation.Furthermore, the probability-wave equation is a partialdifferential equation, and its solution is more difficult thanthe solution of the evolution equation which is an ordinarydifferential equation.Only the Wiener integral is necessary to integrate theLangevin equation. But instead of applying it, textbookskeep busy with the ill-defined Ito Integral that attempted togeneralize the Wiener Integral.We solve the second order Langevin equation for theharmonic oscillator, driven by B(,) ζ t .To date, all random differential equations were assumed tobe driven by B(,) ζ t . There is no reason to believe that none2

Gauge Institute Journal,H. Vic Dannonare driven by Poisson Processes.Thus, we develop here the theory of Random differentialequations driven by Poisson Processes.Keywords: Ito Integral, Ito Process, Ito Formula, StochasticIntegration, Infinitesimal, Infinite-Hyper-real, Hyper-real,Calculus, Limit, Continuity, Derivative, Integral, DeltaFunction, Random Variable, Random Process, RandomSignal, Stochastic Process, Stochastic Calculus, ProbabilityDistribution, Bernoulli Random Variables, BinomialDistribution, Gaussian, Normal, Expectation, Variance,Random Walk, Poisson Process, Random DifferentialEquations, Stochastic Differential Equations2000 Mathematics Subject Classification 26E35; 26E30;26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;46S20; 97I40; 97I30.3

Gauge Institute Journal,H. Vic DannonContentsIntroduction1. Hyper-real Line2. Hyper-real Function3. Integral of a Hyper-real Function4. Delta Function5. Random Walk B(,)ζ t6. Integration sums of f () t with respect to B(,)ζ t .7. Evolution of Linear Oscillator XB (,) ζ t Driven by B(,) ζ t8. Linear Oscillator Process XB (,) ζ t Driven by B(,) ζ t9. RC Linear Oscillator XB (,) ζ t Driven by Thermal NoiseVoltage B(,) ζ t10. RL Linear Oscillator XB (,) ζ t Driven by Thermal NoiseVoltage B(,) ζ t11. Evolution of Harmonic Oscillator XB (,) ζ t Driven byB(,) ζ t12. Harmonic Oscillator Process XB (,) ζ t Driven by B(,) ζ t13. RLC Harmonic Oscillator XB (,) ζ t Driven by ThermalNoise Voltage B(,) ζ t4

Gauge Institute Journal,H. Vic Dannon14. Poisson Process P(,)ζ t15. Integration sums of f () t with respect to P(,)ζ t .16. Evolution of Linear oscillator XP (,) ζ t due to Shot NoiseVoltage P(,) ζ t17. Linear Oscillator Process XP (,) ζ t due to Shot NoiseVoltage P(,) ζ t18. RC Linear Oscillator XP (,) ζ t driven by Shot NoiseVoltage P(,) ζ t19. RL Linear Oscillator XP (,) ζ t driven by Shot NoiseVoltage P(,) ζ t20. Evolution of Harmonic Oscillator XP (,) ζ t due to ShotNoise Voltage P(,) ζ t21. Harmonic Oscillator Process XP (,) ζ t due to Shot NoiseVoltage P(,) ζ t22. RLC Harmonic Oscillator XP (,) ζ t driven by Shot NoiseVoltage P(,) ζ tReferences5

Gauge Institute Journal,H. Vic DannonIntroduction0.1 The Evolution of Random ProcessesA Random Differential Equation is the evolution equation ofa Random Process X(,)ζ t , driven by B (,) ζ t , where B (,) ζ t isRandom Walk, or by P (,) ζ t , where P (,) ζ t is a PoissonProcess.For instance, the Langevin equation for the linear oscillatoris a first order evolution equation for the linear oscillatorprocess, driven by B(,) ζ t . To date, due to the confusionsurrounding Random Differential Equations, only thatequation was integrated and given a Random Processsolution.We solve here the second order Langevin equation for theharmonic oscillator, driven by Random walk. Our method ofVariation of parameters may be applied to Langevinequation of any order.0.2 Probability-waves associated with X(,)ζ t , and thetime-evolution of X(,)ζ t6

Gauge Institute Journal,H. Vic DannonRandom Processes were described by the probability-waveequations associated with them: The Diffusion Equation forRandom Walk, and the differential-difference equation forthe Poisson Process. [Dan5].Integrating the probability-wave equation associated withthe random processX(,)ζ tis equivalent to integrating theRandom Differential Equation for the time-evolution of theProcess X(,)ζ t .But the evolution equation solution is superior to the setup,and solution of the probability-wave equation:While the probability-wave equation setup is instructive,[Dan6], it involves Conditional Probabilities that for higherorder equations are beyond comprehension, reminding ofthe Monty Hall Problem [Rosenhouse].Consequently, it will be difficult to detect an error in thesetup of the probability-wave equation.Furthermore, the probability-wave equation is a partialdifferential equation, and its solution is more difficult thanthe solution of the evolution equation which is an ordinarydifferential equation.7

Gauge Institute Journal,H. Vic Dannon0.3 The Wiener Integral, and the Ito IntegralTo integrate the equations of the time-evolution of randomwalk, Wiener defined the integral of f () t with respect to therandom walk B(,)ζ t .Wiener’s Integral enables the solution of the first orderLangevin equation for the linear oscillator, which isequivalent to the Focker-Planck equation for the probabilitywave.Only the Wiener integral is necessary to integrate theLangevin equation. But instead of applying it, authors keepbusy with the ill-defined Ito Integral that intendedunnecessarily to generalize the Wiener Integral.In [Dan5], we defined in Infinitesimal Calculus the Integralt=b∫ f () tdB(, ζ t), where f () t is integrable hyper-real function,t=aandB(,)ζ tis a Random Walk.In [Dan7] we showed that f () t may not be replaced with aHyper-real Random Process f (,) ζ t , and that the Ito integralthat purports to do that is ill-defined, and does not exist.8

Gauge Institute Journal,H. Vic Dannon0.4 Evolution Equations Driven by Poisson ProcessTo date, all random differential equations are for RandomProcessesXB (,) ζ t, driven by Random Walk Process.But Shot Noise Processes are driven by Poisson Processes.We develop here the theory of Random differential equationsfor ProcessesXP (,) ζ tdriven by Poisson Processes.9

Gauge Institute Journal,H. Vic Dannon1.Hyper-real LineThe minimal domain and range, needed for the definitionand analysis of a hyper-real function, is the hyper-real line.Each real number α can be represented by a Cauchysequence of rational numbers, ( r , r , r ,...) so that r → α .1 2 3The constant sequence ( ααα , , ,...) is a constant hyper-real.In [Dan2] we established that,1. Any totally ordered set of positive, monotonicallyndecreasing to zero sequencesfamily of infinitesimal hyper-reals.( ι1, ι2, ι3,...)constitutes a2. The infinitesimals are smaller than any real number,yet strictly greater than zero.1 1 13. Their reciprocals ( , , ,...ι 1ι 2ι 3) are the infinite hyperreals.4. The infinite hyper-reals are greater than any realnumber, yet strictly smaller than infinity.10

Gauge Institute Journal,H. Vic Dannon5. The infinite hyper-reals with negative signs aresmaller than any real number, yet strictly greater than−∞.6. The sum of a real number with an infinitesimal is anon-constant hyper-real.7. The Hyper-reals are the totality of constant hyperreals,a family of infinitesimals, a family ofinfinitesimals with negative sign, a family of infinitehyper-reals, a family of infinite hyper-reals withnegative sign, and non-constant hyper-reals.8. The hyper-reals are totally ordered, and aligned alonga line: the Hyper-real Line.9. That line includes the real numbers separated by thenon-constant hyper-reals. Each real number is thecenter of an interval of hyper-reals, that includes noother real number.10. In particular, zero is separated from any positivereal by the infinitesimals, and from any negative realby the infinitesimals with negative signs, −dx .11. Zero is not an infinitesimal, because zero is notstrictly greater than zero.11

Gauge Institute Journal,H. Vic Dannon12. We do not add infinity to the hyper-real line.13. The infinitesimals, the infinitesimals withnegative signs, the infinite hyper-reals, and the infinitehyper-reals with negative signs are semi-groups withrespect to addition. Neither set includes zero. ∞14. The hyper-real line is embedded in , and isnot homeomorphic to the real line. There is no bicontinuousone-one mapping from the hyper-real ontothe real line.15. In particular, there are no points on the real linethat can be assigned uniquely to the infinitesimalhyper-reals, or to the infinite hyper-reals, or to the nonconstanthyper-reals.16. No neighbourhood of a hyper-real is nhomeomorphic to an ball. Therefore, the hyperrealline is not a manifold.17. The hyper-real line is totally ordered like a line,but it is not spanned by one element, and it is not onedimensional.12

Gauge Institute Journal,H. Vic Dannon2.Hyper-real Function2.1 Definition of a hyper-real functionf () x is a hyper-real function, iff it is from the hyper-realsinto the hyper-reals.This means that any number in the domain, or in the rangeof a hyper-real f () x is either one of the followingrealreal + infinitesimalreal – infinitesimalinfinitesimalinfinitesimal with negative signinfinite hyper-realinfinite hyper-real with negative signClearly,2.2 Every function from the reals into the reals is a hyperrealfunction.13

Gauge Institute Journal,H. Vic Dannon3.Integral of Hyper-real FunctionIn [Dan3], we defined the integral of a Hyper-real Function.Let f () x be a hyper-real function on the interval [ ab] , .The interval may not be bounded.f () x may take infinite hyper-real values, and need not bebounded.At eacha≤x≤b,there is a rectangle with basedx[ x − , x + 2], height f () x ,dx2and areaf ( xdx. )We form the Integration Sum of all the areas for the x ’sthat start at x = a, and end at x = b,∑ f ( xdx ) .x∈[ a, b]If for any infinitesimal dx , the Integration Sum has thesame hyper-real value, then f () x is integrable over theinterval [ ab] , .14

Gauge Institute Journal,H. Vic DannonThen, we call the Integration Sum the integral of f () x fromx = a, to x = b, and denote it byx=b∫ f ( xdx ) .x=aIf the hyper-real is infinite, then it is the integral over [, ab] ,If the hyper-real is finite,x=b∫ fxdx ( ) = real part of the hyper-real . x=a3.1 The countability of the Integration SumIn [Dan1], we established the equality of all positiveinfinities:We proved that the number of the Natural Numbers,Card , equals the number of Real Numbers,2 Card Card = , and we have2 Card2Card Card = ( Card) = .... = 2 = 2 = ... ≡ ∞.In particular, we demonstrated that the real numbers maybe well-ordered.15

Gauge Institute Journal,H. Vic DannonConsequently, there are countably many real numbers in theinterval[,] ab, and the Integration Sum has countably manyterms.While we do not sequence the real numbers in the interval,the summation takes place over countably many f ( xdx. )The Lower Integral is the Integration Sum where f ( x ) isreplacedby its lowest value on each interval3.2∑x∈[ a, b]⎛⎜⎝dx dx2 2[ x − , x + ]⎞inf f ( t)dx⎠⎟x− ≤t≤ x+dx dx2 2The Upper Integral is the Integration Sum where f ( x ) isreplaced by its largest value on each interval3.3∑x∈[ a, b]⎛⎜⎝⎞ sup f ( t)dx⎠⎟dx dx2 2x− ≤t≤ x+[ x − , x + ]dx dx2 2If the integral is a finite hyper-real, we have16

Gauge Institute Journal,H. Vic Dannon3.4 A hyper-real function has a finite integral if and only ifits upper integral and its lower integral are finite, and differby an infinitesimal.17

Gauge Institute Journal,H. Vic Dannon4.Delta FunctionIn [Dan4], we defined the Delta Function, and established itsproperties1. The Delta Function is a hyper-real function definedfrom the hyper-real line into the set of two hyper-reals⎧⎪ 1 ⎫⎨0, ⎪⎬. The hyper-real 0 is the sequence 0, 0, 0,... .⎪⎩dx⎭⎪ The infinite hyper-real 1dxdepends on our choice ofdx .2. We will usually choose the family of infinitesimals thatis spanned by the sequences1n , 12n,1n3,… It is asemigroup with respect to vector addition, and includesall the scalar multiples of the generating sequencesthat are non-zero. That is, the family includesinfinitesimals with negative sign. Therefore,1dxwillmean the sequence n . Alternatively, we may choose18

Gauge Institute Journal,H. Vic Dannonthe family spanned by the sequences12 n ,13 n ,14 n ,… Then, 1dxwill mean the sequence2 n . Once we determined the basic infinitesimal dx ,we will use it in the Infinite Riemann Sum that definesan Integral in Infinitesimal Calculus.3. The Delta Function is strictly smaller than ∞4. We define,1χ δ ( x) ≡ dx ( ),dx xdx⎡ ⎤ ,⎢−⎣ 2 2 ⎥⎦whereχ ⎡⎢−⎣dx,dx2 2⎧ dx dx1, x ∈ ⎡−, ⎤( x)= ⎪ ⎢ 2 2 ⎥⎨ ⎣ ⎦ .⎪⎪ 0, otherwise⎩⎤⎥⎦5. Hence, for x < 0 , δ ( x) = 0 at fordxx =− , δ( x)jumps from 0 to2dx dx⎢ ⎣,2 2 ⎥ ⎦ , 1( x)x ∈ ⎡−⎤δ = .dx1dx , at x = 0 ,δ (0) =1dx atdxx = , δ( x)drops from2 for x > 0 , δ ( x) = 0.1dxto 0.19

Gauge Institute Journal,H. Vic Dannon xδ ( x) = 06. If dx =1, ( x) = 1 1( x),2 1 1( x),3 1 1( x )...n[ − , ] [ − , ] [ − , ]δ χ χ χ2 2 4 4 6 67. If dx =2,n8. If dx =1,n1 2 3δ ( x) = , , ,...2 2 22 cosh x 2 cosh 2x 2 cosh 3x−x −2x −3x[0, ∞) [0, ∞) [0, ∞)δ( x) = e χ ,2 e χ , 3 e χ ,...x =∞∫9. δ( xdx ) = 1.x =−∞k =∞1 −ik( ξ−x)10. δξ ( − x)= e2π∫ dkk =−∞20

Gauge Institute Journal,H. Vic Dannon5.Random Walk B(,)ζ tThe Random Walk of small particles in fluid is named afterBrown, who first observed it, Brownian Motion. It modelsother processes, such as the fluctuations of a stock price.In a volume of fluid, the path of a particle is in any directionin the volume, and of variable size5.1 Bernoulli Random Variables of the WalkWe restrict the Walk here to the line, in uniforminfinitesimal size steps dx :To the left, with probability1p = ,2or to the right, with probability21

Gauge Institute Journal,H. Vic Dannonq= 1 − p = .12At time t , afterN infinitesimal time intervals dt ,N =tdt, is an infinite hyper-real,the particle is at the pointx .At the i th step we define the Bernoulli Random Variable,Bi (right step)Bi (left step)where i = 1,2,..., N .= dx , ζ1= right step .=−dx, ζ2= left step .Pr( B = dx)= ,i12Pr( B =− dx)= ,iEB [ ] = dx⋅ + ( −dx) ⋅ = 0,i121 12 22 2 1 2 1i2 22E[ B ] = ( dx) ⋅ + ( −dx) ⋅ = ( dx)Var[ Bi] = E[ Bi ] − ( E[ B ]) 0i= 2( dx)2 205.2 The Random WalkB t B B B( ζ , ) =1+2+ ... +N22

Gauge Institute Journal,H. Vic Dannonis a Random Process withEB [ ( ζ , t)] = 0,Var[ B( ζ , t)] = N( dx).2Proof: Since theB iare independent,EB [ ( ζ , t)] = EB [ ] + ... + EB [ N] = 0 10 0Var[ B( ζ , t)] = Var[ B ] + ... + Var[ BN] = N( dx). 1( dx ) ( dx )2 225.3 B(, ζ t + dt) −B(,)ζ t is a Bernoulli Random VariableB i25.4 ( dx) = (2 D)dt ⇒ Random Walk is ContinuousProof:E[{ B( ζ, t + dt) − B( ζ, t)} ] =2= Var[ B(, ζ t + dt) − B(,)] ζ t + ( E[ B(, ζ t + dt) −B(,)]) ζ t 2 ,BiBiwhereB iis a Bernoulli Random Variable,= Var[ B ] + ( [ ]) = (2 ) . iE BiD dt2( dx ) = (2 D)dt0223

Gauge Institute Journal,H. Vic Dannon5.5 If2( dx) = (2 D)dtThen The Derivative of Random Walk is1B = Bdt i,where (1)Bi= B(, ζ t0+ dt) −B(, ζ t0), is a BernoulliRandom Variable.(2) EB [ ] = 0,(3) Var[ B] = 2 Dδ( t ),0Proof:(1) For each t = t 0, we need to find a Random SignalB(, ζ t ), so that for any dt ,0⎡2 ⎤⎡B(, ζ t0 + dt) −B(, ζ t0)⎤E − B ( ζ, t0) = infinitesimal ,⎢⎢dt⎥⎣⎦ ⎥⎣⎦SinceB(, ζ t0+ dt) −B(, ζ t0), is a Bernoulli Random VariableB i,⎡ 2 ⎤ ⎡2 ⎤XBt ( , dt) B( , t)BiE ⎧⎪ + − ζ⎫ ⎧ ⎫B(,) ζ t ⎨− ⎪⎬ = E ⎪ −B⎪⎨⎬⎢⎪⎩ dt⎪⎭ ⎥ ⎢⎪⎩dt⎣ ⎦ ⎪⎭⎥⎣ ⎦Therefore, at timet = t 0, the Random Variable1dtB i,24

Gauge Institute Journal,H. Vic Dannonis the derivative of the Random Walk B(, ζ t ).0(2)(3)1EB [ ] = EB [ ] 0dt i= .Var[ B] = E[ B ] −( E[ B])02 20===12( dt)2( dx) 1dtdt2D2iEB [ ] ,2( dx )(2 D) δ( t ).025

Gauge Institute Journal,H. Vic Dannon6.Integration Sums ofrespect to B(,)ζ t .ft ()withLet f () t be a hyper-real function on the bounded timeinterval [ ab ,].f () t need not be bounded.t=bIn [Dan5], we defined the Integral ∫ f () tdB(, ζ t):t=aAt each a ≤ t ≤b, there is a Bernoulli Random VariabledB(,) ζ t = B(, ζ t + dt) − B(,) ζ t = B (,) ζ t = B (,) ζ t dt.We form the Integration SumFor any dt ,t= b t=b∑f () tdB(, ζ t) = ∑ f() tBi(, ζ t).t= a t=a(1) the First Moment of the Integration Sum is⎡t= b ⎤ t=bE f() t Bi(, ζ t) = f() t E[ B (, ζ t)] = 0⎢⎣t= a ⎥⎦t=a∑ ∑ .(2) the Second Moment of the Integration sum isii026

Gauge Institute Journal,H. Vic Dannon⎡2t= b ⎤ t= b τ=b⎛ ⎞ ⎡⎛ ⎞⎛⎞⎤E f() t Bi(, ζ t) E f() t Bi(, ζ t) f( τ) Bj(, ζ τ)=⎟∑ ∑ ∑⎜ ⎝t= a ⎠⎟⎢⎝⎜t= a ⎠⎝ ⎟⎜⎣τ=a⎢⎥⎟⎠⎥⎣⎦⎦t= b τ=b= ∑∑t= a τ=af ()( tfτ) EB [ (, ζ τ) B(, ζ t)]Since the Bernoulli Random Variables are independent,j2 2jζτiζ =iζ =EB [ ( , ) B( , t)] EB [ ( , t)] ( dx)ionly for t= τ . Then,⎡2t= b ⎤ t=b⎛⎞2 2E f() t Bi(, ζ t) = f ()( t dx)⎜t= a ⎟t=a⎢⎝⎠⎣⎥⎦(2 Ddt )∑ ∑ ,t=b= 2 D∫ f ( t) dt,t=a2assuming2( dx) = (2 D)dt , where D is the Drift coefficient ofthe Random Walk, and assuming that the hyper-realf () t isintegrable.Thus, for any dt , the Integration Sum is a unique welldefinedhyper-real Random VariableIBt=b() ζ = ∫ f() t dB(,)ζ t .t=a27

Gauge Institute Journal,H. Vic Dannon7.Evolution of Linear OscillatorDriven by B(,) ζ t7.1 The Evolution Equation of Linear OscillatorProcess driven by Random Force B(,) ζ tA particle of mass m , affected by a Random ForceB(,) ζ t , isdrifting in a fluid with viscosity γ , with random velocityv(,)ζ t .The balance of forces on the particle isdv(,)ζ tm + γv(,) ζ t = B (,) ζ tdt1dv(,) ζ t = B(,) ζ tdt −γ v(,)ζ tdtm mα dB(,)ζ t βinfinitesimal diffusioninfinitesimal driftThus, the 1 st order Evolution Equation for the LinearOscillator isdv(,) ζ t = αdB(,) ζ t − βv(,)ζ t dt ,v(,) ζ t = αB(,) ζ t −βv(,)ζ t28

Gauge Institute Journal,H. Vic Dannon7.2 The Integrating Factor Solution X (,) ζ tτ=t−βt−β( t−τ)Bτ=0X (,) ζ t = e X (,0) ζ + α∫ e dB (, ζ τ )BBProof:Multiplyingby its integrating factor e βt ,dX (,) ζ t = α dB (,) ζ t − β X (,) ζ t dtBβt βt βtBe dX (,) ζ t = α e dB (,) ζ t − β e X (,) ζ t dt ,βt βt βte dXB(,) ζ t + βe XB(,) ζ t dt = αe dB(,)ζ t ,βtd{ e XB(,) ζ t }Summing over time,βτβt{ B }βtd e X (,) ζ t = αe dB(,)ζ t .τ= tτ=t∑βτ{ }∑d e XB(, ζτ) = α e dB(, ζτ),τ = 0 τ=0τ=tβτ{ e XB(, ζτ)}τ=0βtτ=te X(,) ζ t − X(,0) ζ = α∑ e τ dB(, ζ τ).τ=0Since e is integrable, by section 6, e dB(, ζτ)sums upτ=tBβτβτ=t∑τ=0βτto the Random Variable IB() ζ = ∫ e dB(, ζ τ), andτ=0βτB29

Gauge Institute Journal,H. Vic Dannonτ=te βtX B(,) t X B(,0) e βτζ − ζ = α∫dB (, ζ τ),τ=tτ=0τ=0X − t− ( t−)B(,) ζ t = e β X B(,0) ζ = α∫e β τ dB (, ζ τ).We obtain the same solution for the 1 storder LangevinEquation by the Variation of Parameters Method.7.3 The Variation of Parameters SolutionProof:−βt−β( t−τ)XB(,) ζ t = XB(,0) ζ e + α e dB(, ζ τ) ∫X (,) ζ t τ=0homogeneousτ=tXparticular(,) ζ tdX (,) ζ t = α dB (,) ζ t − β X (,) ζ t dt ,BX (,) ζ t + β X (,) ζ t = α B (,) ζ t .BBTo find the general solution for the homogeneous equation,X (,) ζ t + β X (,) ζ t = 0 ,Bwe substitute in it the separation of variables solutionhom (,) ζBBX t = A e .mtmt() ζmtA() ζ me + βA() ζ e = 0,30

Gauge Institute Journal,H. Vic Dannon( m + β) A( ζ) emt = 0,m+ β = 0 ,mTo find a particular solution of=− β ,hom (,) () tX ζ t = A ζ e −β.X (,) ζ t + β X (,) ζ t = α B (,) ζ t ,Bwe substitute in it the Variation of Parameter SolutionB− tX (,) ζ t = A(,) ζ t e βBt t tAe −β −β −β− Aβe + βAe = αB(,) ζ t ,0tA = αe β B(,) ζ ttAdt = αe β B(,) ζ t dt τ=tA τ=0A(,) ζ t −A(,0)ζdA dB(,)ζ tτ=tβτ(, ζτ) = α∫e dB(, ζτ),τ=0Therefore,τ=tA(,) ζ t = A(,0) ζ + α∫ e dB(, ζ τ),τ=0βτ− tX (,) ζ t = A(,) ζ t e βB31

Gauge Institute Journal,H. Vic Dannonτ=t−βt−βtβτ= A(,0) ζ e + αe ∫ e dB(, ζ τ)τ=0τ=t−βt−β( t−τ)∫= X(,0) ζ e + α e dB(, ζ τ).Xhomogeneous( ζ, t)τ=0X ( ζ, t)particular32

Gauge Institute Journal,H. Vic Dannon8.Linear Oscillator ProcessXB (,) ζ tDriven by8.1 The Normal Random ProcessB(,) ζ tτ=tX t( t )B(,) t e −β X B(,0) e −β −τζ = ζ + α∫dB (, ζ τ)τ=0is Normally Distributed withMean−βtEX [ ( ζ, t )] = e EX [ ( ζ,0)],BBand Variance222Var[ X ttB( , t )] e − βVar X B( , 0) D α(1 e − βζ = ⎡ ζ ⎤⎣ ⎦+ − ).βProof: SincedB(, ζτ)is Normally Distributed, so isτ=tτ=0−β( t−τ)IB() ζ = ∫ e dB(, ζ τ). Hence, XB (,) ζ t has Normaldistribution.⎡ τ= t ⎤ τ=tβτβτE[ IB()] ζ = E e dB(, ζ τ) ∑ = ∑ e E[ dB(, ζ τ )] = 0.⎢⎣τ= 0 ⎥⎦τ=0Bi033

Gauge Institute Journal,H. Vic Dannon−βtEX [B( ζ, t)] = Ee [ XB( ζ,0)] + E[ αe IB( ζ)] ,−βt−βt−βte E[ XB( ζ,0)] αe E[ IB( ζ)]−βtζ= e E[ X B( ,0)].2 2Var[ IB()] ζ = E ⎡IB () ζ ⎤ −( E ⎡IB())ζ ⎤⎣⎢ ⎦⎥ ⎣ ⎦,τ=t= ∑ eτ=02βτ2τ=tτ=0E[ B ]i20( dx) = (2 D)dτ= 2D∑ e d = 2 D ( e −1).2βτ1 2βtτ2β−βt−βtVar[ XB( ζ, t)] = Var ⎡e XB( ζ, 0) ⎤ + Var ⎡αe IB( ζ)⎤⎢⎣ ⎥⎦ ⎢⎣⎥ ⎦−2βt2 2 te Var ⎡ βXB( ζ,0) ⎤ −α e Var ⎡IB( ζ)⎤⎣ ⎦ ⎣ ⎦−2βt22 te Var XB( , 0) Dα − β= ⎡ ζ ⎤⎣ ⎦+ (1 −e).,β08.2 The Linear Oscillator Equilibrium StateAt Equilibrium, t = infinite hyper-real Θ,EX [B( ζ, Θ)]≈Var[ X DB( ζ, Θ)] ≈ D α = .β γ20m34

Gauge Institute Journal,H. Vic Dannon9.RC Linear Oscillator ProcessqB (,) ζ t Driven by B(,) ζ t9.1 The Evolution Equation of the RC LinearOscillator ProcessqB (,) ζ t Driven by B(,) ζ tThermal Noise in the Resistor R , generates Random VoltageB (,) ζ t on the Resistor R , that drives a random currentB (,) dqB (,) ζ ti ζ t = through the Circuit.dtqB (,) ζ t is the random charge on the Capacitor CThe balance of voltages on the circuit components isdq B(,) ζ t q B(,)ζR + t = B (,) ζ t ,dt C35

Gauge Institute Journal,H. Vic Dannon1 1dqB(,) ζ t = B(,) ζ tdt−qB(,)ζ tdt.R RCαdB(,)ζ tβ9.2 The Random Charge Process q (,) ζ tτ=t−1t1 − 1 ( t−τ)RCRCBζBζRτ=0q (,) t = e q (,0) + ∫ e dB(, ζ τ )Proof: By section 8,τ=t−βt−β( t−τ)Bτ=0q (,) ζ t = e q (,0) ζ + α∫ e dB (, ζ τ ),Bτ=t−1t1 − 1 ( t−τ)RCRCBζRτ=0= e q (,0) + ∫ e dB (, ζ τ ).B9.3 The Normal Distribution of the Random Chargeτ=t−1t1 − 1 ( t−τ)RCRCBζBζRτ=0q (,) t = e q (,0) + ∫ e dB(, ζ τ )is Normally Distributed withMean1t RCEq [ ( ζ, t )] = e Eq [ ( ζ,0)],B−Band Variance−2 t−2Bζ = ⎡Bζ ⎤⎣ ⎦+ −R1 1RCCRCVar[ q ( , t)] e Var q ( , 0) D (1 e )t36

Gauge Institute Journal,H. Vic DannonProof: By 8.1,1t RC−βt−B B BEq [ (,)] ζ t = e Eq [ (,0)] ζ = e Eq [ (,0)] ζ .222Var[ q ttB( , t )] e − βVar q B( , 0) D α(1 e − βζ = ⎡ ζ ⎤⎣ ⎦+ − )−2 t−2⎡Bζ ⎤⎣ ⎦ R1 1RCCRC= e Var q ( , 0) + D (1 − e ).βt9.4 The Random Charge Steady StateAt the Steady State, t = infinite hyper-real Θ,Eq [B( ζ, Θ)] ≈ 0.Var[ q ( ζ, Θ)] ≈ =DC.BD α β2R9.5 The Charge Noise Energy at the Steady StateProof:1 12 2C2BkT = E[ q ( ζ, Θ)] =D,where k is Boltzmann Constant,and T is the absolute Temperature.2B2REq [ ( ζ, Θ )] = Θ + Θ =2C12{Var[ q ( , )] ( [ ( , )]) }D2 Bζ E qCBζ 2R0DCR.37

Gauge Institute Journal,H. Vic Dannon10.RL Linear Oscillator ProcessiB (,) ζ t Driven by B(,) ζ t10.1 The Evolution Equation of the RL LinearOscillator ProcessiB (,) ζ t Driven by B(,) ζ tThermal Noise in the Resistor R , generates Random VoltageB (,) ζ t on the Resistor R , that drives a random currentiB (,) ζ tthrough the Circuit.diB (,) ζ tL is the Random Voltage on the Solenoid L .dtThe balance of voltages on the circuit components isdiB(,)ζ tL + iB(,) ζ t R = B (,) ζ t ,dt38

Gauge Institute Journal,H. Vic Dannon1RdiB(,) ζ t = B(,) ζ tdt−iB(,)ζ tdt.L LαdB(,)ζ tβ10.2 The Random Current Process i (,) ζ tProof: By 7.2,τ=t−RtR1 − ( t−τ)LLBζBζLτ=0i (,) t = e i (,0) + ∫ e dB(, ζ τ )τ=t−βt−β( t−τ)Bτ=0i (,) ζ t = e i (,0) ζ + α∫ e dB (, ζ τ ),Bτ=t−RtR1 − ( t−τ)LLBζLτ=0= e i (,0) + ∫ e dB (, ζ τ ).B10.3 The Normal Distribution of i (,) ζ tτ=tRR− t1 − ( t−τ)LLBζBζLτ=0i (,) t = e i (,0) + ∫ e dB(, ζ τ )is Normally Distributed withBMeanRt LEi [ ( ζ, t )] = e Ei [ ( ζ,0)],B−Band Variance39

Gauge Institute Journal,H. Vic DannonR−2 t1 −2LBζ = ⎡Bζ ⎤⎣ ⎦+ −RLVar[ i ( , t)] e Var i ( , 0) D (1 eL)Proof: By 8.1,Ei [ (,)] ζ t = e Ei [ (,0)] ζ = e Ei [ (,0)].Rt L− β t−B B B ζ222Var[ i ttB( , t )] e − βVar i B( , 0) D α(1 e − βζ = ⎡ ζ ⎤⎣ ⎦+ − )R−2 t1 −2L ⎡Bζ ⎤⎣ ⎦ RL= e Var i ( , 0) + D (1 − e L).βRtRt10.4 The Random Current Steady StateAt the Steady State, t = infinite hyper-real Θ,Ei [B( ζ, Θ)] ≈ 0.Var[ i ( ζ, Θ)] ≈ =D.BD α β2RL10.5 The Current Noise Energy at the Steady StateProof:1 12 22BkT = E[ i ( ζ, Θ)] L =D,where k is Boltzmann Constant,and T is the absolute Temperature.1 2 12Ei [ (, )] {Var[ (, )] ( [ (, )])}D2 Bζ Θ L= i2 Bζ Θ + EiB ζ Θ L= 2R0DRL2R.40

Gauge Institute Journal,H. Vic Dannon11.Evolution of HarmonicOscillator Driven by B(,) ζ t11.1 Evolution equation of Harmonic Oscillator drivenby Random Force B(,) ζ tA particle of mass m , affected by a Random ForceB(,) ζ t , isdrifting in a fluid with viscosityγ , and Hooke’s LawConstant k , with random velocity x(,) ζ t .The balance of forces on the particle isdx(,) ζ tm + γx (,) ζ t + kx(,) ζ t = B(,) ζ t ,dt1dx(,) ζ t = B(,) ζ tdt − γ x(,) ζ tdt −k x(,)ζ tdtm m mα dB( ζ, t)β 2 ωinfinitesimal diffusioninfinitesimal driftinfinitesimal Hooke'sThus, the evolution Equation for the Harmonic Oscillatordriven by a Random WalkB(,)ζ tisordx (,) ζ t = α dB (,) ζ t −β x (,) ζ t dt −ω x B(,)ζ t d t.Bx 2 (,) ζ t = α B (,) ζ t −β x (,) ζ t −ω x (,) ζ t .BBB2B41

Gauge Institute Journal,H. Vic Dannon11.2 Variation of Parameters Solutionfor the Harmonic Oscillator ProcessdX 2(,) ζ t = α dB (,) ζ t −β X (,) ζ t dt −ω X (,) ζ t d t,BBBis− β tκt1 21 12 2XB(,) ζ t = e { A(,0) ζ e + A (,0) ζ e }Xhomogeneous(,) ζ t−κ1t 2τ=tα m1( t−τ) m2( t−τ)+ { e −e } dB ζτκ∫( , ),τ=0X (,) ζ tparticularprovided2 2κ = β − 4ω> 0,m =− β + κ,1 11 2 2Proof:m =− β − κ1 12 2 2dX 2(,) ζ t = α dB (,) ζ t −β X (,) ζ t dt −ω X (,) ζ t d t,BX (,) ζ t + βX (,) ζ t + ω X (,) ζ t = αB (,) ζ t .2B B BTo find the general solution for the homogeneous equation,2X (,) ζ t + βX (,) ζ t + ω X (,) ζ t = 0 ,B B Bwe substitute in it the separation of variables solutionhom (,) ζA() ζ m e + βA() ζ me + ω A() ζ e = 0,BX t = A e .mt() ζ2 mt mt 2 mtB42

Gauge Institute Journal,H. Vic Dannon2 2( m + βm + ω ) A( ζ) emt = 0,mm2 2βmω 0+ + = ,1 11 2 22 2=− β +β −4ω,1 12 2 2κ>02 2m =− β − β −4ωmthom 1 2X (,) ζ t A () ζ e A () ζ eTo find a particular solution ofmt=1+2.X (,) ζ t + βX (,) ζ t + ω X (,) ζ t = αB (,) ζ t ,2B B Bwe substitute in it the Variation of Parameter Solutionso thatThenmt1 2X (,) ζ t A (,) ζ t e A (,) ζ t eBmt=1+2, mt mt.Ae1 21+ Ae2= 0mt1mt2mt1mtζ ⎡ 2 ⎤XB(,)t = Ame1 1+ Am2 2e + ⎢Ae 1+ Ae⎣2 ⎥⎦, (,) 2,mt mt 2 mt 2 mt1 1 2 2 1 1 2 2X ζ t = Am e1+ Am e2+ Am e1+ Am eB2αB = X + βX + ω X= + + +mt mt 2 mt 2 mt1 1 2 2 1 1 2 2Ame1Am e2m Ae1m Ae02+mt1mt21 1βAme2 2+ βAme++2 mt1 2 mtAe21ω Ae2+ ω +43

Gauge Institute Journal,H. Vic Dannon mt1mtζ ζ2+= A (,) t m e + A (,) t m e1 1 2 22 2 mt 2 2+ ( m1)21+ βm1 + ω ) Ae1+ ( m2 + βm2 + ω Ae2. 0 0This yields two equations for A 1, and A 2,In matrix form, mt mt,Ae1 21+ Ae2= 0 .mtmt1 1 2 2Ame1+ Am e2= αB⎡ ⎤⎡ ⎤ ⎡ ⎤.⎣⎦⎣⎢ ⎦⎥ ⎣ ⎦mt1mt2e e A10 ⎥⎢ mt1mtme2 A1me⎥⎢ ⎥=2 2αB⎢ ⎥ ⎢ ⎥mtAmt0 e2mtαB m2mt22e −αBe αB= = = eκ1 mt1mt2( m1+m2)te e ( m2 − m1)emt1mtme21me −κ2−mt1,mtAdt α −1= e 1 Bdt κdA1dB( ζ, t)τ=tτ= t α −mτA1 = τ=0κ∫τ=0A (,) ζ t −A(,0) ζ(, ζτ) e1dB (, ζτ)1 144

Gauge Institute Journal,H. Vic Dannonτ=tα −mτA1 t A1e dBκ(,) ζ = (,0) ζ +1∫ (, ζ τ )τ=0Amte10me1αB mtαBe 1αB= = = − eκmt12 mt1mt2( m1+m2)te e ( m2 − m1)emt1mtme21me −κ2−mt2mtAdt α −2=− e 2 Bdt κdA2dB( ζ, t)τ=tτ= t α −mτA2 ζτ =− τ=0κ∫τ=0A (,) ζ t −A(,0) ζ(, ) e2dB (, )2 2ζτTherefore,τ=tα −m2τA2(,) ζ t = A2(,0) ζ − e dB(, )κ∫ ζ τ ,mt1 2X (,) ζ t = A (,) ζ t e + A (,) ζ t eBmt1 2τ=0τ=tmt α1m1 ( t−τ)= A1(,0) ζ e + e dB(, ζ )κ∫ τ +τ=0τ=tmt α2m2 ( t−τ)+ A2(,0) ζ e − e dB(, ζ )κ∫ ττ=045

Gauge Institute Journal,H. Vic Dannonτ=tmt mt α m( t−τ) m ( t−τ)1ζ2ζκτ=0= A(,0) e1+ A (,0) e2+ { e1e2∫ − } dB(, ζ τ ),−tκt1ζ21β12 2= e { A( ,0) e + A ( ζ,0) e }τ=tτ=0−1κt2α m1( t−τ) m2( t−τ)+ { e −e } dB ζτκ∫( , ).11.3 The Time-Rate Random Process X(,) ζ t−1 t1X (,) t e β−1 ζ =2{ m A(,0) ζ e2+ m A (,0) ζ eκt2}Bτ=tκt1 1 2 2α+ −κm1( t−τ) m2( t−τ)∫ { me1me2} dB( ζτ , ),τ=0provided2 2κ = β − 4ω> 0,m =− β + κ,1 11 2 2m =− β − κ1 12 2 2Proof: From the proof of 11.2,X(,) ζ t = A (,) ζ t m e + A (,) ζ t m eBmt1mt21 1 2 2− βtκt1 1ζ2 21 12 2= e { m A( ,0) e + m A ( ζ,0)eτ=tα+ −κ−κt12}m1( t−τ) m2( t−τ)∫ { me1me2} dB( ζτ , ).τ=046

Gauge Institute Journal,H. Vic Dannon12.Harmonic Oscillator ProcessDriven by B(,) ζ t12.1 The Normal Distribution of the HarmonicOscillator Processκt1 21 12 2X (,) ζ t = e { A(,0) ζ e + A (,0) ζ e }B− β tτ=tα+ −κ−1κt2m1( t−τ) m2( t−τ)∫ { e e } dB( ζτ , )τ=0is Normally Distributed withMeanmt1 2EX [ (,)] ζ t EA [ (,0)] ζ e EA [ (,0)] ζ eB=1+2,and Variance−( ) t( ) tVar[ XB( , t)] e β − κ −Var A1( , 0) e β +ζ = ⎡ ζ ⎤ + κ Var ⎡A2( ζ, 0) ⎤⎣ ⎦ ⎣ ⎦+α ⎧⎪+ − + −κ⎪⎩β − κ β β + κ2 −( β−κ) t −βt − ( β+κ)t2 e e eD ⎪⎨2mt⎫⎪⎬+⎪⎭2 2 22D α β −+ωβω β − 4ω2 2 2,provided2 2κ = β − 4ω> 0,47

Gauge Institute Journal,H. Vic Dannonm =− β + κ,1 11 2 2m =− β − κ.1 12 2 2Proof: SincedB(, ζτ)is Normally Distributed, so isBτ=tm ( t−τ) m ( t−τ)∫ B(,)ζ tτ=0I e e dB() ζ = {1−2} (, ζ )Normal Distribution.⎡ τ=t⎤m1( t−τ) m2( t−τ)EI [B( ζ)] = E { e e } dB( ζ, τ)∑ −,⎢⎣τ= 0⎥⎦τ=t∑m ( t−τ) m ( t−τ)τ . Hence, X has= { e1−e 2} EdB [ ( ζτ , )] = 0.τ=0Bimt1 2EXBt EA e A e E I κ B[ (,)] [ (,0)1(,0)2 αζ = ζ + ζ ] + [ ()] ζmt 1 mt 2{ EA [1( ,0)] e + EA [2( ,0)] e }0mtζ ζ αEI [B( ζ)]κ mt1ζ2= EA [ ( ,0)] e + EA [ ( ζ,0)]emt1 2.02 2Var[ IB()] ζ = E ⎡IB () ζ ⎤ −( E ⎡IB())ζ ⎤⎣⎢ ⎦⎥ ⎣ ⎦,τ=tm1( t−τ) m2( t−τ) 2 2= ∑ { e −e } E[ Bi]τ=002( dx) = (2 D)dτ,48

Gauge Institute Journal,H. Vic Dannonτ=t∫2 m ( t−τ) −β( t−τ)2 m ( t−τ)= 2 D { e1− e + e2} dτ.τ=0⎧2mt1 −βt2mte 1 e 1 e2 ⎫2D⎪ − − −1= ⎨ + + ⎪⎬⎪ 2m1 β 2m⎩2 ⎪⎭⎡ mtmt ⎤ ⎡ ⎤1 2κ BVar[ (,)] Var{ (,0)1(,0)2αX ζ t A ζ e A ζ e } Var I () ζB= ⎢ + ⎥ + ⎢⎣ ⎥⎣⎦ ⎦2mt1 2mte Var ⎡A 21( ζ,0) ⎤ e Var ⎡A2( ζ,0)⎤ α⎣ ⎦+⎣ ⎦Var ⎡I( )2 Bζ ⎤⎣ ⎦κ⎡ ⎤ ⎡⎣ ⎦ ⎣2mt2mt1 2= e1Var A( ζ, 0) + e2Var A ( ζ, 0) +2 2mt1 βt2mtα ⎧−e 1 e 1 e2 ⎫2D⎪ − − −1+ ⎨ + + ⎪⎬2κ⎪ 2m1 β 2m⎩2 ⎪⎭−( β−κ) t− ( β+κ)t= e Var ⎡A ⎤1( ζ, 0) ⎤ e Var ⎡⎣ ⎦+⎣A2( ζ, 0)⎦+α ⎧⎪+ − + −κ⎪⎩β − κ β β + κ2 −( β−κ) t −βt− ( β+κ)t2 e e eD ⎪⎨22α ⎧1 1 1 ⎫+ 2D⎪⎨ − + ⎪⎬.2κ⎪⎩β − κ β β + κ ⎪⎭2β1 β 1− = −2 2 β 2β −κ 2ωβ2 2 2α β −2ωDβω β −4ω2 2 2⎤⎦⎫⎪⎬⎪⎭212.2 The Harmonic Process Equilibrium StateAt Equilibrium, t = infinite hyper-real Θ,49

Gauge Institute Journal,H. Vic DannonEX [B( ζ, Θ)]≈ 0,2 2 22Var[ XB( , )] D α β −ζ Θ ≈ ω > 0 ,2 2 2βω β − 4ω2 2provided β − 4ω> 0.Proof:m Θ1 2EX [ (, ζ Θ )] = EA [ (,0)] ζ e + EA [ (,0)] ζ eBm Θ1 21 2 2 1 2 2( β β 4 ω ) ( β β 4 ω2 2− − − Θ − + − ) Θ1ζ2ζ= EA [ ( , 0)] e + EA [ ( , 0)] e.Since we assume2β > 4ω2, then,2 2β − β − 4ω> 0, and2 2β + β − 4ω> 0, andee12 2− ( β− β − 4 ω ) Θ2 ≈ 012 2− ( β+ β − 4 ω ) Θ2 ≈ 0Hence, EX [ ( , )] 0.B ζ Θ ≈ −( β−κ) Θ − ( β+ κ)ΘVar[ XB( ζ, Θ )] = e Var ⎡A1( ζ, 0) ⎤ + e Var ⎡A2( ζ, 0) ⎤⎣ ⎦ ⎣+ ⎦≈0, since β− κ> 0 ≈ 0, since β+ κ>02 ( β κ) β ( β κ)α ⎧− − Θ − Θ − + Θe e e ⎫+ 2D⎪⎨− + − ⎪⎬+2κ⎪⎩β − κ β β + κ ⎪⎭2 2 22D α β −+ωβω β − 4ω2 2 2≈0≈2 2 22D α β − ωβω β − 4ω2 2 22 2> 0 , since β > 0, and β − 4ω> 0.50

Gauge Institute Journal,H. Vic Dannon12.3 Normal Distribution of the Process X(,) ζ t mt1mtζ = ζ + ζ2+X (,) t A (,) t m e A (,) t m eB1 1 2 2τ=tα+ −κm1( t−τ) m2( t−τ)∫ { me1me2} dB( ζτ , )τ=0is Normally Distributed withMeanmt[ ( , )] [ ( , 0)]1mtEX ζ t = mEA ζ e + mEA [ ( ζ, 0)] e2,B1 1 2 2and VarianceVar[ 2 ( )X( , )] − β−κtBζ t = m1 e Var ⎡A1( ζ , 0) ⎤⎣ ⎦+2 ( β κ)t+ m2 e− + Var ⎡ ⎣A2( ζ, 0) ⎤ ⎦+β− κ −( β−κ) t βt β κ ( β κ)t{ 4ω − + − +}2 2+αD − e + e − e + Dκ22 β2B2α ,2βprovided2 2κ = β − 4ω> 0,m =− β + κ,1 11 2 2m =− β − κ.1 12 2 2Proof: SincedB(, ζτ)is Normally Distributed, so isBτ=tm ( t−τ) m ( t−τ)∫ 1 2. Hence, XB(,)ζ tτ=0J m e m e dB() ζ = {1−2} (, ζ τ)has Normal Distribution.51

Gauge Institute Journal,H. Vic Dannon⎡ τ=t⎤m1( t−τ) m2( t−τ)EJ [B( ζ)] = E { me1me2} dB( ζ, )∑−τ,⎢⎣τ= 0⎥⎦τ=t∑m ( t−τ) m ( t−τ)= { me1 21−me 2} EdB [ ( ζτ , )] = 0.τ=0Bimt1mtEX [ ( , )] [2αBζ t = EmA1 1( ζ,0) e + mA2 2( ζ,0) e ] + E[2 JB( ζ)] κmt 1 mt 2{ mEA1[1( ,0)] e + mEA2[2( ,0)] e }ζ ζ α2 EJ [B( ζ)]κ mt1 1ζ2 2= mEA [ ( ,0)] e + mEA [ ( ζ,0)]e1 20mt.02 2Var[ JB( ζ)] = E ⎡JB ( ζ) ⎤ ( E ⎡JB( ζ) ⎤⎢ ⎥ −⎣ ⎦) ,⎣ ⎦ τ=tm1( t−τ) m2( t−τ) 2 2= ∑ { me1−me 2} EB [i]τ=0τ=t∫02( dx) = (2 D)dτ2 2 m ( t−τ) −β( t−τ) 2 2 m ( t−τ)1 1 2 2= 2 D { m e1− 2 m m e + m e2} dτ.τ=0⎧2mt1 −βt2mt⎫e 1 e 1 e22D⎪−−−1= ⎨m1 + 2m 1m 2+ m ⎪2 ⎬2 β22⎪⎩ω⎪⎭,⎡ mtmt ⎤ ⎡ ⎤1 1 2 2κ BVar[ X (,)] ζ t Var{ m A (,0) ζ e1m A (,0) ζ e2} VarαJ () ζB= ⎢ + ⎥ + ⎢⎣ ⎥⎣⎦ ⎦2 2mt1 2 2mtm 21e Var ⎡A1( ζ,0) ⎤ m2 e Var ⎡A2( ζ,0) ⎤ α⎣ ⎦+⎣ ⎦ Var ⎡J( )2 Bζ ⎤⎣ ⎦κ252

Gauge Institute Journal,H. Vic Dannon2 2mt1 2 2mt= m21e Var ⎡A1( ζ, 0) ⎤ m2 e Var ⎡⎣ ⎦+⎣A2( ζ, 0) ⎤⎦+22mt1 βt2mtα ⎧−e 122 e 1 e ⎫2D⎪−−−1+ ⎨m2 1+ 2ω+ m ⎪2 ⎬κ⎪ 2 β2⎩⎪⎭2 −( β−κ) t2 − ( β+κ)t= m ⎤1e Var ⎡A1( ζ, 0) ⎤ m2 e Var ⎡⎣ ⎦+⎣A2( ζ, 0)⎦+1 κtω 1−κ{ ( β κ) 4 ( β κ)}α2 −βtD e e 2t+ − − + − + e2κ2 β 21 1{ ( β κ) − 4 + ( β + κ)}2 2+ D α − ω .2κ 2 β 2D2 2β −4ωβ2αβ12.4 The Time-Rate Process Equilibrium StateAt Equilibrium, t = infinite hyper-real Θ,Proof:EX [ ( ζ, Θ)]≈ 0,BVar[ X B( ζ, Θ)] ≈ D α > 0 ,β2 2provided β − 4ω> 0.EX [ (, ζ Θ )] = mEA [ (,0)] ζ e + mEA [ (,0)] ζ eBm Θ1 1 2 21 22m Θ1 2 2 1 2 2( β β 4 ω ) ( β β 4 ω2 2− − − Θ − + − ) Θ1[1( ζ,0)] 2[2( ζ,0)]= mEA e + mEA e.53

Gauge Institute Journal,H. Vic DannonSince we assume2β > 4ω2 , then,2 2β − β − 4ω> 0, and2 2β + β − 4ω> 0, andee12 2− ( β− β − 4 ω ) Θ2 ≈ 012 2− ( β+ β − 4 ω ) Θ2 ≈ 0Hence, EX [ ( ζ , Θ B)] ≈ 0.2 ( )Var[ X− β−κΘB( ζ, Θ )] = m1 e Var ⎡A1( ζ, 0) ⎤⎣ ⎦+≈ 0, since β+ κ>0≈0, since β− κ>02 − ( β+ κ)Θ+ m2 e Var ⎡A2( ζ, 0) ⎤⎣ ⎦+ 2 2α ⎧β − κ −( β−κ) Θ ω −βΘ β + κ ⎫− ( β+ κ)Θ+ 2D⎪⎨− e + 2 e − e ⎪⎬+2κ⎪⎩4 β 4 ⎪⎭2+ D α β≈02≈ D α > 0 ,βsince β > 0.54

Gauge Institute Journal,H. Vic Dannon13.RLC Harmonic OscillatorDriven by Thermal NoiseVoltage B(,) ζ t13.1 The Evolution Equation of RLC HarmonicProcess driven by Thermal Noise Voltage B(,) ζ tThermal Noise in the Resistor R , generates Random VoltageB (,) ζ t on the Resistor R , that drives a Random CurrentB (,) dqB (,) ζ ti ζ t = through the Circuit.dti (,) ζ t R = q (,) ζ t R is the Random Voltage on R .BBdiB(,)ζ tLdt= Lq (,) ζ t is the Random Voltage on L .B55

Gauge Institute Journal,H. Vic DannonqB (,) ζ tCis the Random Voltage on C .The balance of voltages on the circuit components isLq t + q t R + q t = Bζ t ,1B(,) ζB(,) ζB(,) ζ (,)Cq 1 R(,) (,) (,)1Bζ t =B(,) B ζ t −L q ζ t −L q tLC B ζ .α β ω213.2 The Thermal Noise Charge q (,) ζ tBR R 2 C 2−4 LC R 2 C 2−4tt−LC2L 2LC 2LCq (,) ζ t = e { A(,0) ζ e + A (,0) ζ e } +B−1 2tτ= t ⎧ R RC 2 2−4LC ⎫ R RC 2 2 4LC( t τ) ⎧ −⎫−⎪ ⎨ − ⎪⎬ − − ⎪⎨ + ⎪⎬( t−τ)2L2LC2L2LC⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭C+ { ee}2 2∫ −dB ( ζτ , )RC − 4LCτ=0Proof: By 11.2, the Harmonic Oscillator evolution equationdq B(,) ζ t = α dB (,) ζ t −β q B(,) ζ t dt −ω q B(,)ζ t dt .is solved by the Processq (,) ζ t = e { A(,0) ζ e + A (,0) ζ e } +B− 1βt1κt−1κt2 2 21 2τ=tα+ −κm1( t−τ) m2( t−τ)∫ { e e } dB( ζτ , ),τ=02provided2 2κ = β − 4ω> 0,56

Gauge Institute Journal,H. Vic DannonSubstitutingthen,1α = ,LRLm =− β + κ,1 11 2 2m =− β − κ.221 12 2 2R 2 1β = , ω = ,L LC1 1 2 24 RC 4LC LCκ = − = − LC,mm2 2−4R R C L1 2L2LC=− + C ,2 2−4R R C L1 2L2LC=− − C ,− ⎧⎪qB( ζ, t) = e ⎨A1( ζ,0) e + A2( ζ,0)e⎪⎩R R 2 C 2−4 LC R 2 C 2−4tLC2L ⎪t−2LC 2LCt⎫⎪⎬+⎪⎭τ= t ⎧ R RC 2 2−4LC ⎫ R RC 2 2 4LC( t τ) ⎧ −⎫−⎪ ⎨ − ⎪⎬ − − ⎪⎨ + ⎪⎬( t−τ)2L2LC2L2LC⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭C+ { ee}2 2∫ −dB ( ζτ , ).RC − 4LCτ=013.3 Normal Distribution of the Thermal Noise Charge− ⎧⎪qB( ζ, t) = e ⎨A1( ζ,0) e + A2( ζ,0)e⎪⎩R R 2 C 2−4 LC R 2 C 2−4tLC2L ⎪t−2LC 2LCt⎫⎪⎬+⎪⎭τ= t ⎧ R RC 2 2−4LC ⎫ R RC 2 2 4LC( t τ) ⎧ −⎫−⎪ ⎨ − ⎪⎬ − − ⎪⎨ + ⎪⎬( t−τ)2L2LC2L2LC⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭C+ { ee}2 2∫ −dB ( ζτ , )RC − 4LCτ=0is Normally Distributed with57

Gauge Institute Journal,H. Vic DannonMeanR 2 2 4 2 2tR C − LCt−R C −4LC2L2LCEq [ ( ζ, t)] = e { EA [ ( ζ,0)] e + EA [ ( ζ,0)] e2LC},B−and Variance1 2tR 2 2 4 2 2tR C − LCt−R C −4LCLLCVar[ q ( ζ, t)] = e {Var[ A( ζ,0)] e + Var[ A ( ζ,0)] eLC}B−1 2t2 2 4 2 2 42 −Rt⎧R C − LCt−R C − LCt2DC LCeL eLC1 eLC+ ⎪⎨+ −2 2RC − 4LC 2 2 2 2RC R C 4LC RC⎪− + − RC + R C − 4LC⎩⎫⎪⎬⎪⎭+2CRCD R 2RC− 2L.− 4LProof: By 12.1,−mt1 2Eq [ ( ζ, t)] = EA [ ( ζ,0)] e + EA [ ( ζ,0)]eBmt1 2R R2C2−4LC R2C2−4LC−2L2LC2LCt t t1ζ2ζ= e { E[ A(,0)] e + E[ A (,0)] e }.−( ) ( )Var[ ( , )] ttq t e β − κ −Var A1( , 0) e β +ζ = ⎡ ζ ⎤ + κ Var ⎡A2( ζ, 0) ⎤⎣ ⎦ ⎣ ⎦+α ⎧e e e+ 2D⎪⎨− + −2κ⎪⎩β − κ β β + κ2 −( β−κ) t −βt − ( β+κ)t⎫⎪⎬+⎪⎭2 2 22D α β −+ωβω β − 4ω2 2 2,−R R 2 C 2−4 LC R 2 C 2−4tt−LCL LC L= e {Var[ A( ζ, 0)] e + Var[ A ( ζ, 0)] eC} +1 2t58

Gauge Institute Journal,H. Vic DannonR R2C2−4LC R2C2−4LC2 − t ⎧t− t2DC LCeL eLC1 eLC+ ⎨⎪+ −2 2RC − 4LC 2 2 2 2− RC + R C − 4LC RC RC + R C − 4LC⎪⎩⎫⎬⎪⎪⎭+2CRCD R 2RC− 2L.− 4L13.4 The Thermal Noise Charge Steady StateAt the Steady State, t = infinite hyper-real Θ,Eq [B( ζ, Θ)] ≈ 0.2CRC−2LqBζ Θ ≈ D R 2RC−LVar[ ( , )]4.13.5 The Noise Energy of qB (,) ζ t at the Steady State21 1 2kT = E[ q ( , )]2 2CBζ Θ =2D R C − 2L,2R RC−4Lwhere k is Boltzmann Constant,and T is the absolute Temperature.Proof:1 2 12Eq2 Bζ Θ = qC2CBζ Θ + EqBζ Θ 2CRC−2L0[ (, )] {Var[ (, )] ( [ (, )])}22R 2RC−4D R 2RCD RC−2L= .L−4L59

Gauge Institute Journal,H. Vic Dannon13.6 The Thermal Noise Current i (,) ζ tBR2 22 2R C − 4 LCt2LR R C −4LCtζ2LC +−Bζ = − +2L 2LCq(,) t e ( ) A (,0) e1−R2 22 242 ( 4) ( ,0)R C − LCttL R R C − LC−ζ2LC +− e + A e2 22L2LC2 2R R C −4LC2L2LC2τ=tτ=0R R2C2−4 LC2L2LC−[ − ]( t−τ)−C( − ) ∫ edB(ζτ , )+RC −4LC2 22 2R R C −4LC2L2LCτ=tτ=0R R2C2− 4 LC2L2LC− [ + ]( t−τ)C+ ( + ) ∫ edB ( ζτ , ).RC −4LCProof: By 11.3,κt1 1 2 21 β12 2q(,) ζ t = e { m A(,0) ζ e + m A (,0) ζ e }B−tτ=tα+ −κ−1κt2m1( t−τ) m2( t−τ)∫ { me1me2} dB( ζτ , ),τ=0−R2 22 242 ( 4) ( ,0)R C − LCttL R R C − LCζ2LC += e − + A e2L2LC1−R2 22 242 ( 4) ( ,0)R C − LCttL R R C − LC−ζ2LC +− e + A e2 22L2LC2 22R R C −4LC2L2LCτ=tτ=0R R2C2− 4 LC2L2LC−[ − ]( t−τ)C− ( − ) ∫ edB(ζτ , )+RC −4LC2 22 2R R C −4LC2L2LCτ=tτ=0R R2C2− 4 LC2L2LC− [ + ]( t−τ)C+ ( + ) edB ( ζτ , ).RC −4LC∫60

Gauge Institute Journal,H. Vic Dannon13.7 Normal Distribution of Thermal Noise Currenti (,) ζ t = q (,) ζ tBBis Normally Distributed withMeanR2 22 2R C − 4 LCt2LR R C −4LCtζ2LC +−Bζ = − +2L 2LCEi [ ( , t )] e ( ) EA [ ( ,0)] e1−and VarianceR2 22 242 ( 4) [ ( ,0)]R C − LCttL R R C − LC−ζ2LC ,− e + E A e2L2LC2R 2 22 2 44 2R C −tLCL R R C LCLC− − tBζ = − +⎡2L 2LC⎣Var[ i ( , t )] e ( ) e Var A1( ζ , 0) ⎤⎦+−R 2 22 2 44 2R C −tLCL R R C LC −LC+ e ( + − ) e Var ⎡A2( ζ,0)⎤⎣ ⎦+2L2LCtR− t22 2L C − −2 2RC −4LCRC R C 4LC2LCR2C2− 4 LCLC− e D e +−Rt L+ e D+2C2 2RC −4LC4RCtProof: By 12.3,−RLt2C2 2RC −4LCR2C2− 4 LCLCRC +2 2R C −4LC− t D2LC2RL− e D e +mtmt1 1 2 2i (,) ζ t A (,) ζ t m e A (,) ζ t m eB=1+2+τ=tα+ −κm1( t−τ) m2( t−τ)∫ { me1me2} dB( ζτ , )τ=0is Normally Distributed withMean61

Gauge Institute Journal,H. Vic Dannonmt1 1 2 2Ei [ ( ζ, t)] mEA [ ( ζ,0)] e mEA [ ( ζ,0)]eBmt=1+2,−R2 22 242 ( 4) [ ( ,0)]R C − LCttL R R C − LC2LC += e − + E A ζ e−2L2LCR2 22 2R C − 4 LCt2LR R C −4LC−2L− e ( + ) E[ A ( ζ ,0)] e2L2LCand VarianceVar[ ( , )] 2 −( β−κ)i tBζ t = m1 e Var ⎡A1( ζ , 0) ⎤⎣ ⎦+2 − ( β+ κ)t+ m2 e Var ⎡A2( ζ, 0) ⎤⎣ ⎦+β− κ −( β−κ) t ω −βt β+κ − ( β+κ)t{ }2 224 β421tC.+α2D − e + 2 e − e + Dκ2α ,2βR 2 22 2 44 2R C −tLCL R R C LCLC= e − ( − + − ) e Var⎡A1( ζ,0)⎤⎣ ⎦+2L2LCt−R 2 22 2 44 2R C −tLCL R R C LC −LC+ e ( + − ) e Var ⎡A2( ζ,0)⎤⎣ ⎦+2L2LCtR− t22 2L C − −2 2RC −4LCRC R C 4LC2LCR2C2− 4 LCLC− e D e +−Rt L+ e D+2C2 2RC −4LC4RCt−R2 222 2R C − 4 LCtL C RC + R C −4LC−LC2 2RC −4LCt D2LC2RL− e D e + .13.8 The Thermal Noise Current Steady StateAt the Steady State, t = infinite hyper-real Θ,Ei [B( ζ, Θ)] ≈ 0.62

Gauge Institute Journal,H. Vic DannonDVar[ iB( ζ, Θ)]≈ .2RL13.9 Thermal Noise Energy of iB (,) ζ t at Steady State2 DkT = E[ iB( ζ, Θ )] L = ,4R1 12 2where k is Boltzmann Constant,and T is the absolute Temperature.Proof:1 2 122 Bζ Θ =2 Bζ Θ +Bζ Θ D02RLEi [ (, )] L {Var[ i (, )] ( Ei [ (, )])} LD= .4R13.10 RLC Harmonic Oscillator Thermal Noise Energy2D ⎧R C 2L1⎫⎪ −⎨ + ⎪⎬.R ⎪⎩RC−4L⎪⎭2 2 263

Gauge Institute Journal,H. Vic Dannon14.Poisson Process P(,)ζ tThe arrival at rate λ , of radioactive particles at a counter ismodeled by the Poisson Process. It models other processes,such as the arrival of phone calls at rate λ , to an operator.14.1 The Bernoulli Random Variables of the ProcessWe assume thatan arrival probability in time dt isp= λdtand no arrival probability in time dt is,q= 1 − λdt.At fixed time t , afterN infinitesimal time intervals dt ,N =tdt, is an infinite hyper-real,there areandk arrivals,k is a finite hyper-realN− k no arrivals,64

Gauge Institute Journal,H. Vic DannonN− k is an infinite Hyper-realAt the i th step we define the Bernoulli Random Variable,where i = 1,2,..., N .Pi(arrival) = 1, ζ1= arrivalPi (no-arrival) = 0 , ζ2= no-arrivalPr( P = 1) = p = λdt,iPr( P = 0) = q = 1 − λdt,iE[ P ] = 1⋅ λdt + 0 ⋅(1 − λdt)= λdt,i2 2 2iE[ P ] = 1 ⋅ λdt + 0 ⋅(1 − λdt)= λdt,2 2iiλdtλdtVar[ P] = E[ P ] − ( E[ P]),i= λdt (1 −λdt)≈ λdt.≈114.2 The Binomial Distribution of the ProcessP( ζ , t) = P + P + ... + PN1 2is a Random Process withEP [ ( ζ, t)]= λt,Var[ P( ζ, t)]= λt,Proof: Since theP iare independent,65

Gauge Institute Journal,H. Vic DannonEP [ ( ζ, t)] = EP [ 1 ] + ... + EP [ ]N= λNdtλdtλdtVar[ P( ζ, t)] = Var[ P1] + ... + Var[ PN] ≈ λNdt ≈λdt≈λdttt14.3 P(, ζ t + dt) −P(,)ζ t is a Bernoulli Random Variable14.4 Poisson Process is ContinuousProof:E[{ P( ζ, t + dt) − P( ζ, t)} ] =2= Var[ P( ζ, t + dt) − P( ζ, t)] + ( E[ P( ζ, t + dt) −P( ζ, t)]) 2 ,PiPiwhereX iis a Bernoulli Random Variable,= Var[ Pi] + ( E [ Pi]) = infinitesimal .≈λdt 2λdt14.5 The Derivative of the Poisson process1P = Pdt i,where (1)Pi= P(, ζ t0+ dt) −P(, ζ t0), is a BernoulliRandom Variable.66

Gauge Institute Journal,H. Vic Dannon(2) EP [ ] = λ ,(3) Var[ P] = λδ( t0)Proof:(1) For each t = t 0, we need to find a Random SignalP(, ζ t ), so that for any dt ,0⎡2 ⎤⎡P(, ζ t0 + dt) −P(, ζ t0)⎤E − P ( ζ, t0) = infinitesimal,⎢⎢dt⎥⎣⎦ ⎥⎣⎦SinceP(, ζ t0+ dt) −P(, ζ t0), is a Bernoulli Random VariableP i,⎡ 2 ⎤ ⎡2 ⎤P(, t dt) P(,)tPiE ⎪⎧ ζ + − ζ⎫ ⎧ ⎫P(,) ζ t ⎨− ⎪⎬ = E ⎪ −P⎪⎨⎬⎢⎪⎩ dt⎪⎭ ⎥ ⎢⎪⎩dt ⎪⎭⎣ ⎦⎥⎣ ⎦Therefore, at timet = t 0, the Random Variable1dtis the derivative of the Random Walk P(, ζ t ).P i,0(2)(3)1EP [ ] = EP [ ]dt i= λ .λdtVar[ P] = E[ P ] −( E[ P])2 2λ67

Gauge Institute Journal,H. Vic Dannon1= EP [ ]2( ) i− λdt1= λdt= λδ( t ),02 22 2λdt+λ ( dt)By [Dan4].68

Gauge Institute Journal,H. Vic Dannon15.Integration sums offt ()withrespect toP(,)ζ tLet f () t be a hyper-real function on the bounded timeinterval [ ab ,].f () t need not be bounded.At each a ≤ t ≤b, there is a Bernoulli Random VariabledP(,) ζ t = P(, ζ t + dt) − P(,) ζ t = P(,) ζ t = P (,) ζ t dt.We form the Integration SumFor any dt ,t= b t= b t=b∑ ∑ ∑ ζf () tdP(, ζ t) = ftP () (, ζ t) = ftP () (, tdt )it= a t= a t=a(1) the First Moment of the Integration Sum is⎡t= b ⎤ t=bt=bE f () t P(,) ζ t dt = f () t E[ P∑ ∑ (,)] ζ t dt = λ ∫ f () t dt ,⎢⎣t= a ⎥⎦t= a t=aλiassuming f () t integrable.(2) the Second Moment of the Integration sum is69

Gauge Institute Journal,H. Vic Dannon⎡2t= b ⎤ t= b τ=b⎛ ⎞ ⎡⎛ ⎞⎛⎞⎤E f() t Pi(, ζ t) E f() t Pi(, ζ t) f( τ) Pj(, ζ τ)∑ =∑ ∑⎟⎜ ⎝t= a ⎠⎟⎢⎝⎜t= a ⎠⎝ ⎟⎜⎣τ=a⎢⎥⎟⎠⎥⎣⎦⎦t= b τ=b= ∑∑t= a τ=af ()( tfτ) EP [ (, ζ τ) P(, ζ t)]Since the Bernoulli Random Variables are independent,only for t2j i iEP [ ( ζτ , ) P( ζ, t)] = EP [ ( ζ, t)] = λdt(1 + λdt)= τ . Then,⎡⎤⎛E f t B t = f t⎜⎢⎝⎠⎣⎥⎦ji≈12t= b ⎞ t=b2∑ ()i(, ζ ) λ ()∑ dt ,⎥t= a ⎟ t=at=b= λ ∫ f2 () tdt,t=aassuming f () t integrable.Thus, assuming f () t integrable, for any dt , the IntegrationSum is a unique well-defined hyper-real Random VariableI (). ζ We call I () ζ the integral of f () t , with respect toPPP(,)ζ t from x = a, to x = b, and denote it byt=b∫ f () tdP(, ζ t).t=a70

Gauge Institute Journal,H. Vic Dannon16.Evolution of Linear Oscillatordue to Shot Noise Voltage P(,) ζ t16.1 Shot Noise VoltageSince the electron charge is−191.6 × 10 Coulomb , a current of310 − 6.24 × 1015Ampere has aboutelectrons.the number of electrons will fluctuate by millions ofelectrons per second, and the fluctuations modeled by aPoisson ProcessP(,) ζ t .P(,)ζ t , will generate a Shot Noise VoltageThis noise which is independent of the temperature, and thefrequency, will be added to the Thermal Noise that isproportional to the temperature, and to the Flicker Noisewhich spectral density is inversely proportional to thefrequency.At higher temperatures, shot noise is negligible compared tothe Thermal Noise, and at lower frequencies, it is negligiblecompared to the Flicker noise.71

Gauge Institute Journal,H. Vic DannonBut at low temperatures, and high frequencies, Shot Noisemay be the main noise.16.2 Evolution Equation of Linear Oscillator ProcessXP (,) ζ tdriven by Shot Noise VoltageThe balance of voltages in a linear oscillator circuit due tothe Shot Noise Voltage Component isdX (,) ζ t = α dP (,) ζ t − β X (,) ζ t dt .PP16.3 Integrating Factor Solutionτ=t−βt−β( t−τ)Pτ=0X P(,) ζ t = e X (,0) ζ + α∫e dP (, ζ τ )Proof: following the proof of 7.2.We obtain the same solution for the linear oscillatorevolution equation by the Variation of Parameters Method.72

Gauge Institute Journal,H. Vic Dannon17.Linear Oscillator ProcessXP (,) ζ tVoltagedue to Shot NoiseP(,) ζ t17.1 The Poisson Distribution of X (,) ζ tτ=t−βt−β( t−τ)Pτ=0X P(,) ζ t = e X (,0) ζ + α∫e dP (, ζ τ)is Poisson Distributed withMeanEX [ ( , t )] ( EX t[ ( ,0)] 1) e −ζ = λ + ζ − β ,Pand Varianceα β222 1Var[ X ( , t )] e − tVar X αt( , 0) (1 e −ζ = ⎡ ζ ⎤ + λ − )( − ).PPβ β λ⎣ P ⎦ β2 βProof: Since dP(, ζτ)is Poisson Distributed, so isτ=tI () ζ = ∫ e dP(, ζ τ ). Hence, X (,) ζ t is PoissonPτ=0distributed.−β( t−τ)PP73

Gauge Institute Journal,H. Vic Dannon⎡ τ=t ⎤βτEI [P( ζ)] = E e dP( ζ, )∑ τ,⎢⎣τ= 0 ⎥⎦τ=tβτ= ∑ e E[ dP( ζτ , )]τ=0Piτ=tτ=0λdτβτ λ βt= λ∫ e dτ= ( e − 1).β−βtEX [P( ζ, t)] = Ee [ XP( ζ,0)] + E[ αe IP( ζ)] ,−βt−βt−βte E[ XP( ζ,0)] αe E[ IP( ζ)]−βtλ( eβt−1)β−βtα= e E[ X ( ζ,0)] + λ (1−e )P− t= λ + ( EX [ ( ζ,0)] − 1) e β .α βPβ22Var[ IP( ζ)] = E ⎡ IP ( ζ) ⎤ −E ⎡ IP( ζ)⎣⎢ ⎦⎥ ⎣ ⎤ ⎦,τ=t∑2βτ2λ βt( e −1)βλ= e E[ P ] ( 1)2i− e − ,βτ=0τ=tλdt2βtβτ2λ βt∑ e dτ( e − 1)2= λ −τ=02 2β274

Gauge Institute Journal,H. Vic Dannon2ββt2e −β21( t1) λe(2 2= λ − − 1)ββtλ βte + 1 λ βt= ( e −1) ⎡( e 1)⎤β ⎢ − −⎣ 2 β ⎥⎦βtβt= λ( e − λ1)( e + 1)( − )β2 1e βt2= λ ( − 1)( − λ).β−βt−βtVar[ XP( ζ, t)] = Var ⎡e XP( ζ, 0) ⎤ Var ⎡⎢ ⎥ + ⎢αe IP( ζ)⎤⎣ ⎦ ⎣ ⎥⎦−2βt2 2 te Var ⎡ βXP( ζ,0) ⎤ −α e Var ⎡IP( ζ)⎤⎣ ⎦ ⎣ ⎦β−2 t2α −2 t 1= e Var ⎡X ( , 0) ⎤ + (1 −e)( − ).12β β λ⎣ Pζ⎦λβ2 ββ17.2 The Linear Oscillator Process Steady StateAt Steady State, t = infinite hyper-real Θ,EX [ P( ζ, Θ)]≈αλ β212α λVar[ XP( ζ, Θ)] ≈ λ ( − ).ββ75

Gauge Institute Journal,H. Vic Dannon18.RC Oscillator Process drivenby Shot Noise Voltage P(,) ζ t18.1 Evolution Equation of qP (,) ζ t in RCLinear Oscillator due to Shot Noise Voltage P(,) ζ tCurrent fluctuations in the circuit, generate Shot NoiseVoltage P(,) ζ t that drives a random currentP (,) dqP (,) ζ ti ζ t = through the Circuit.dtqP (,) ζ t is the random charge on the Capacitor CThe balance of voltages due to the shot noise isdq P(,) ζ t q P(,)ζR + t = P (,) ζ t ,dt C76

Gauge Institute Journal,H. Vic Dannon1 1dqP(,) ζ t = P(,) ζ t dt − qP(,)ζ t dt .R RCαdP(,)ζ tβ18.2 The Random Charge Process q (,) ζ tProof: By 16.3,τ=t−1t1 − 1 ( t−τ)RCRCPζPζRτ=0q (,) t = e q (,0) + ∫ e dP(, ζ τ )τ=t−βt−β( t−τ)Pτ=0q (,) ζ t = e q (,0) ζ + α∫ e dP (, ζ τ ),PBy 18.1,1α = ,1β = ,R RCτ=t−1t1 − 1 ( t−τ)RCRCPζRτ=0= e q (,0) + ∫ e dP (, ζ τ ).P18.3 The Poisson Distribution of the Random Chargeτ=t−1t1 − 1 ( t−τ)RCRCPζPζRτ=0q (,) t = e q (,0) + ∫ e dP(, ζ τ )is Poisson Distributed withMeanEq [( ,)] [ (,0)] 1 t RCPζ t λ C Eq Pζ e−= + ⎡ − ⎤⎣⎦,and Variance177

Gauge Institute Journal,H. Vic Dannon1t−1RCCRC−2 2Pζ⎡qP ζ ⎤⎣ ⎦+ D −eRVar[ q ( , t)]= e Var ( , 0) (1 )Proof: By 17.1,−Eq [ ( ζ, t)] = λ + ( Eq [ ( ζ,0)] − 1) e βPα βP− t= λC + ( E[ q ( ζ,0)] − 1) e β .−2βt2α −2βt1 λVar[ qP( ζ, t)] = e Var ⎡qP( ζ, 0) ⎤⎣ ⎦+ λ (1 −e )( − )1t1RCC −RC−2 2 t⎡ ⎤1P ⎣ ⎦ R2P= e Var q ( ζ, 0) + λ (1 − e )( −λR C).βt2βt18.4 The Random Charge q (,) ζ t Steady StatePAt the Steady State,t = infinite hyper-real Θ,Eq [P( ζ , Θ)]≈ λC .Var[ q ( ζ , Θ )] ≈ λC( − λRC) .PR1218.5 The Charge Noise Energy at the Steady Statewherek2λ24REq [ ( ζ, Θ)]= ,Cis Boltzmann Constant,Proof:and T is the absolute Temperature.Eq2ζ[ ( , Θ )] = 1 {Var[(, q ζ Θ )] + ( E[(,q ζ Θ )])}22C2C λC 1R 2( −λRC)λCλ= .4R78

Gauge Institute Journal,H. Vic Dannon19.RL Oscillator Process drivenby Shot Noise VoltageP(,) ζ t19.1 Evolution Equation of Current iP (,) ζ t in RLOscillator due to Shot Noise Voltage P(,) ζ tCurrent fluctuations in the circuit, generate Shot NoiseVoltage P (,) ζ t that drives a random current i (,) ζ t throughdiP the Circuit.(,) ζ tL is the Random Voltage on L .dtThe balance of voltages due to the shot noise isdiP(,)ζ tL + iP(,) ζ t R = P (,) ζ t ,dt1RdiP(,) ζ t = P(,) ζ tdt−iP(,)ζ tdt.L LαdP(,)ζ tβP79

Gauge Institute Journal,H. Vic Dannon19.2 The Current Process iP (,) ζ t due to Shot NoiseProof: By 16.3,τ=tRR− t1 − ( t−τ)LLP=P+L ∫τ=0i (,) ζ t e i (,0) ζ e dP(, ζ τ)τ=t−βt−β( t−τ)Pτ=0i (,) ζ t = e i (,0) ζ + α∫ e dP (, ζ τ),PBy 19.1,1 Rα = , β = ,LLτ=t−RtR1 − ( t−τ)LLPζLτ=0= e i (,0) + ∫ e dP (, ζ τ). 19.3 Poisson Distribution of the Shot Noise Currentτ=tRR− t1 − ( t−τ)LLP L ∫τ=0i (,) ζ t = e i (,0) ζ + e dP(, ζ τ)Pis Poisson Distributed withMeanRt LEi [ ( , t λζ )] = + ( Ei [ ( ζ,0)] − 1) e ,Pand VarianceRR−2 t1 −2LPζ = ⎡Pζ ⎤⎣ ⎦+ −RLVar[ i ( , t)] e Var i ( , 0) D (1 eL)Proof: By 17.1,P−Rt80

Gauge Institute Journal,H. Vic DannonEi [ ( , t )] α ,0)] 1) e −ζ = λ + ( Ei [ ( ζ −βPβPRt Lλ= + ( Ei [ ( ζ,0)] − 1) e .R222Var[ i ( , t )] e − tVar i t( , 0)α(1 e −ζ = ⎡ ζ ⎤ + λ − )(1− )PPβ β λ⎣ P ⎦ β2 βR−2 t 2Lλ − t⎡ ⎤L 1 λL⎣ P ⎦ RL 2 R= e Var i ( i, 0) + (1 −e)( − ).R−t19.4 Shot Noise Current Steady StateAt the Steady State, t = infinite hyper-real Θ,λEi [ ( ζ, Θ)]≈ .PVar[ ( ζ , Θ )] ≈ ( − ) .Rλ 1 λLiP RL 2 R19.5 Shot Noise Current Energy at the Steady State1Ei 2[ ( , )]2 PL λ4Rζ Θ = ,wherekis Boltzmann Constant,and Tis the absolute Temperature.Proof:1 2 12 λEi [ (, )] {Var[ (, )] ( [ (, )])}2 Pζ Θ L= i2 Pζ Θ + EiP ζ Θ L=4R λ(1−λL)λRL 2 R R. 81

Gauge Institute Journal,H. Vic Dannon20.Evolution of HarmonicOscillator driven by Shot NoiseVoltageP(,) ζ t20.1 Evolution Equation of Harmonic Oscillatordriven by Shot NoiseP(,) ζ tThe Evolution Equation for the Harmonic Oscillator drivenby a Poisson Process P(,) ζ t isdx(,) ζ t = αdP(,) ζ t −βx(,) ζ t dt −ω x(,)ζ t dt,2x(,) ζ t = αP(,) ζ t −βx(,) ζ t −ω x(,)ζ t .220.2 Variation of Parameters Solutionfor the Harmonic Oscillator ProcessdX 2P(,) ζ t = α dP (,) ζ t −β X P(,) ζ t dt −ω X P(,)ζ t d t,is−1 β t1 κ t−1κ t2 2 21 2XP(,) ζ t = e { A(,0) ζ e + A (,0) ζ e } +Xhomogeneuos(,) ζ t82

Gauge Institute Journal,H. Vic Dannonτ=tα+ −κτ=0m1( t−τ) m2( t−τ)∫ { e e } dP( ζτ , ),Xζparticular (,) tprovided2 2κ = β − 4ω> 0,m =− β + κ,1 11 2 2m =− β − κ1 12 2 2Proof: Following the proof of 11.2.20.3 The Time-Rate Random Process X(,) ζ t1 βt1t1tX−(,) t e2{ m A(,0) e κ− κζ = ζ2+ m A (,0) ζ e2}Pτ=t1 1 2 2α+ m 1∫ ( t −τ) m2( t−τ){ me1− me2} dP( ζτ , ),κτ=0Pprovided2 2κ = β − 4ω> 0,m =− β + κ,1 11 2 2m =− β − κ1 12 2 2Proof: Following the proof of 11.3.83

Gauge Institute Journal,H. Vic Dannon21.Harmonic Oscillator Processdue to ShotNoise VoltageP(,) ζ t21.1 The Poisson Distribution o f the HarmonicOscillator Process XP (,) ζ t driven by Shot Noiseκt1 21 1 12 2 2X (,) ζ t = e { A(,0) ζ e + A (,0) ζ e }P− βtτ=tα+ −κm1( t−τ) m2( t−τ)∫ { e e } dP( ζτ , )τ=0is Poisson Distributed withMeanmt1 2EX [ ( ζ, t)] = EA [ ( ζ,0)] e + EA [ ( ζ,0)]eP−κtmt1 2and Variancemt1mtα⎧e e2+ λ ⎪⎨ − +κ⎪⎩m1 m2κω2⎫⎪⎬⎪⎭−( β−κ) t− ( β+κ)tVar[ XP( ζ, t)] = e Var ⎡A1( ζ, 0) ⎤ + e Var ⎡A2( ζ, 0) ⎤⎣ ⎦ ⎣ ⎦+2 ( β κ) t βt ( β κ) t2 2α ⎧− − − − +e e e ⎫2α 2λ⎪⎪ β − ω+ ⎨− + − ⎬+λ2 ( β κ) β β κ2κ− + 2 2⎪⎩⎪⎭2βω β − 4ω +84

Gauge Institute Journal,H. Vic Dannon2−λακ2⎡e⎢⎣mmtmte− +m1 2κ2 21 2 ω⎤⎥⎦2,provided2 2κ = β − 4ω> 0,m =− β + κ,1 11 2 2m =− β − κ.1 12 2 2Proof: Since dP(, ζτ ) is Poisson Distributed, so isτ=tm ( t−τ) m ( t−τ)J () ζ { e e } (, )P∫=1−2dP ζ τ . Hence, (,) isτ= 0Poisson distributed.⎡ τ=t⎤m1( t−τ) m2( t−τ)EJ [P( ζ)] = E { e e } dP( ζ, τ )∑ −,⎢⎣τ= 0⎥⎦τ=t∑m ( t−τ) m ( t−τ)=1 2τ=0{ e −e } E[dP( ζτ , )]Piλdττ=tτ=tmt −m τ mt −mτ∫ ∫ .= λe 1e1dτ −λe 2e2dττ=0 τ=0XPζtλ mt λ= ( e −1) − ( emm1 21 2mt−1)85

Gauge Institute Journal,H. Vic Dannon⎧ mt1mt2 ⎫ ⎧ ⎫e e1 1= λ⎪ ⎨ − ⎪⎬+ λ⎪⎨− + ⎪⎬⎪ m1 m 2 ⎪ m1m⎩ ⎭ ⎪⎩2⎪⎭mte1eλ ⎧mt= ⎪ −⎫ 2⎨⎪⎬ + 2λm m β− κ β+κ⎪⎩ 1 2 ⎪⎭ 1mt2κλ ⎧mt⎪ee ⎫= ⎨ − + ⎬⎪m 2⎪ 1m2ω⎩⎪⎭mt1 21{ −1}κ2 2ωκλω2mtEXPt EA e A e[ (,)] ζ = [ (,0) ζ1+ (,0) ζ2] + E[ α J () ζ ]{ [ 1 21( , mtmtEA ζ 0)] e + EA [2( ζ,0)]e }mt1ζ2= EA [ ( ,0)] e + EA [ ( ζ,0)]emt1 2P καEJ [P( ζ)]κmt1mtα⎧e e2+ λ ⎪⎨ − +κ⎪⎩m1 m2κω2⎫⎪⎬.⎪⎭From the Proof of 12.1,τ=t∑m t m t 2 2{ e1( −τ) e2( −τ)E ⎡JP2 () ζ ⎤⎢ = − } E[ Pi]⎣ ⎥⎦τ=0λdττ=t2 m t−τ−β( t−τ)m ( t−τ)∫ { e 1 ( )2e e 2 } dτ.τ=0= λ− +86

Gauge Institute Journal,H. Vic Dannon=⎧2mt1 −βt2mte 1 e 1 e2 ⎫λ ⎪ − − −1⎨ + + ⎪⎬⎪ 2m1 β 2m⎩2 ⎪⎭⎧ 2mt1 −βt2mt2 ⎫ ⎧ ⎫e e e1 1 1= ⎪⎨ + + ⎪⎬+ ⎪⎨ − + ⎪⎬⎪ 2m1 β 2m⎩2 ⎪⎭⎪⎩β − κ β β + κ⎪⎭Var[ J ( )] E ⎡J ( ) ⎤ ( E ⎡J( ζ) ⎤⎢ ⎥ ⎣ ⎦)⎣ ⎦2 2Pζ =Pζ −P2 2β −2ω2βω⎧⎪2m1t −βte2−1 2⎪2 mt 2 2mt mte e ⎫⎪ β 2ω ⎧ 2 e e κ ⎫= λ⎨ + + ⎬+ λ −λ⎨⎪ − + ⎬⎪2m 2 2⎪ 1β 2m 2 ⎪ 2βω ⎪ m1 m2ω⎩ ⎭ ⎩ ⎪⎭22⎧ 2mt 12mt 2 2 2 mt1mt−βte e e β − 2ω ⎡e e2κ ⎤= λ⎪⎨ + + + −λ− +2m 2 21β 2m2 2βω m1 m2ω⎢⎣⎥⎪⎩⎦2⎫⎪⎬⎪⎭⎡ mtmt ⎤ ⎡ ⎤1 2κ P= ⎢ + ⎥ + ⎢⎣ ⎥⎣ ⎦ ⎦Var[ ( , )] Var { ( , 0)1( , 0)2αX ζ t A ζ e A ζ e } Var J ( ζ)2mt1 2mte Var ⎡A 21( ζ,0) ⎤ e Var ⎡A2( ζ,0)⎤ α⎣ ⎦+⎣ ⎦Var ⎡J( )2 Pζ ⎤⎣ ⎦κ2= e ⎡ ⎤⎣ ⎦+2mt12mtVar A e21( ζ, 0)Var ⎡A2( ζ, 0) ⎤⎣ ⎦+2⎧ 2mt 1 βt2mt 2 2 2 mt1mtα −e e e β − 2ω⎡e e2+ λ⎪⎨ + + + −λ− +2 2m 2κ1β 2m2 2βω⎢⎣m1 m2⎪⎩κω2⎤⎥⎦2⎫⎪⎬⎪⎭−( β−κ) t− ( β+κ)t= e Var ⎡A ⎤1( ζ, 0) ⎤ e Var ⎡⎣ ⎦+⎣A2( ζ, 0)⎦+2 ( β κ) t βt ( β κ) t2 2α ⎧− − − − +e e e ⎫2α 2λ⎪⎪ β − ω+ ⎨− + − ⎬+λ2 ( β κ) β β κ2κ− + 2 2⎪⎩⎪⎭2βω β − 4ω +87

Gauge Institute Journal,H. Vic Dannon−λ2mtmt1 22 α e e κκ⎡⎢⎣m− +m2 21 2 ω⎤⎥⎦2.21.2 The Harmonic Process Equilibrium StateAt Equilibrium, t = infinite hyper-real Θ,αEX [ P( ζ, Θ)]≈ λ ,ω2Proof:2 2 2α ⎧1 β − 2ω1Var[ X P( ζ, Θ)]≈ λ ⎪⎨−λ2 2β2 2ω ⎪⎩β − 4ω ω2 2provided β − 4ω> 0.m Θ1 2EX [ (, ζ Θ )] = EA [ (,0)] ζ e + EA [ (,0)] ζ ePm Θ1 24⎫⎪⎬,⎪⎭m1 mα⎧Θe e2Θ+ λ ⎨⎪ − +κ⎪⎩m1 m2κω2⎫⎪⎬⎪⎭{ EA [ ( ζ,0)]λα}= +1κe2 2−1( β− β − 4 ω ) Θ2+α{ EA [ ( ,0)] }2122 2− ( β+ β −4 ω ) Θα+ ζ − λ e+ λ .κω2Since we assumeβ2 2> 4ω, then,2 2β − β − 4ω> 0, ande1 2 2− β− β − ω Θ≈2 ( 4 ) 02 2β + β − 4ω> 0, ande1 2 2− β+ β − ω Θ≈2 ( 4 ) 088

Gauge Institute Journal,H. Vic DannonHence,EX [ ( ζ, Θ)]≈ λα.Pω2−( β−κ) Θ − ( β+ κ)ΘVar[ XP(, ζ Θ )] = e Var ⎡A1(,0) ζ ⎤ + e Var ⎡A2(,0)ζ ⎤⎣ ⎦ ⎣+ ⎦≈0, since β− κ> 0 ≈ 0, since β+ κ>02 ( β κ) β ( β κ)α ⎧− − Θ − Θ − + Θe e e ⎫+ λ ⎪⎨ + + ⎪⎬+2κ⎪⎩−( β −κ) β − ( β + κ)⎪⎭2 2 22D α β −+ωβω β − 4ω2 2 22 2 1ω4≈02 mt1mt22 α ⎡ee κ ⎤−λ κ2 − +m21m2ω⎢⎣⎥⎦≈λα2≈2 2 2α ⎧1 β 2ω1λ ⎪ −⎨−λω ⎪⎩2 β β − 4ω ω2 2 2 4⎫⎪⎬.⎪⎭21.3 The Poisson Distribution of the Time-RateRandom Process mt1 mt2XP (, ζ t) = A1(,) ζ t m1e + A2(,)ζ t m2e+τ=tα m ( t−τ)m ( t−τ)∫ 1 2κτ=0+ −is Poisson Distributed withζτ{ me1me 2} dP( , )89

Gauge Institute Journal,H. Vic DannonMean⎧mtEX [ ( , t)] mEA1[1( ,0)] α ⎫ ⎧αζ = ⎪⎨ ζ + λ⎪ ⎬e + ⎪⎨mEA 2[2( ζ,0)]− λ ⎫ ⎪ ⎬⎪⎭ e⎪⎩ κ⎪⎭ ⎪⎩κand Variance2 ( ) tVar[ X− β−κ( ζ, t)] = m1 e Var ⎡A1( ζ, 0) ⎤⎣ ⎦+2 ( β κ)t+ m2 e− + Var ⎡ ⎣A2( ζ, 0) ⎤ ⎦+mt1 21 κt1t{ ( ) e 2ωκ( ) e−}2 2 2+ −α λ e −βtκ2− ⎡ − +⎣β − κ + − β + κ + λα4 β 4 2β2 ( β κ) β ( β κ)t2αλ e − − t t2 e − e − +2κ ⎢⎤⎥,⎦,provided2 2κ = β − 4ω> 0,m =− β + κ,1 11 2 2m =− β − κ.1 12 2 2Proof: SincedP(, ζτ)is Poisson Distributed, so isτ= t∫m1( t−τ) m2( t−τ)JP() ζ = { m1e −m2e } dP(, ζ τ). Hence, X P (,) ζ t isτ=0Poisson Distributed.⎡ τ=t⎤m1( t−τ) m2( t−τ)EJ [P( ζ)] = E { me1me2} dP( ζ, τ )∑−,⎢⎣τ= 0⎥⎦90

Gauge Institute Journal,H. Vic Dannonτ=t∑m ( t−τ) m ( t−τ)= { me1 21−me 2} EdP [ ( ζτ , )].τ=0Piλdtτ= tτ=tmt −m τ mt −mτ∫ 1 ∫ 2τ= 0 τ=0= λ−e1m e1dt λe 2m emt= λ e − e(1 2)−mt1 −mt21−e1−emtEX [ 0{ mEA 1 21 [ 1( mtmt,0)] e + mEA }2 [ 2 ( ,0)] e2 dt,mtmt1 1 2 2P κζ ζ αEJ [P( ζ)]κ (,)] ζ t = EmA [ (,0) ζ e1+ mA(, ζ ) e2] + E[ αJ ()] ζmt( 1 mtλ e −e2 )⎧ = ⎪ α ⎫ ⎧mtα ⎫⎨mEA1[1( ζ,0)]+ λ⎬ ⎪e + ⎪⎨m2E[ A2( ζ,0)]− λ⎪⎬e⎪⎩κ⎪⎭ ⎪⎩ κ⎪⎭mt1 2.τ=tE ⎡ 2 ττJ ζ ⎤ m −−= me1( t ) m⎢ ⎥−m e2( t ) 2() E P⎣ ⎦ ∑ {2P1 2} [i]τ=t= λ∫τ=0τ=0λdτ2 2 m1(t−τ) −β( t−τ) 2 2 m ( t−τ)11 2 2{ m e− 2 m m e + m e2} dτ.⎧2mt1 −βt2mt⎫e 1 e 1 e2λ ⎪ − − −1= ⎨m1 + 2m 1m2 + m ⎪2 ⎬2 β22⎪⎩ω⎪⎭91

Gauge Institute Journal,H. Vic Dannon−βt 2 21 κt ω 1 −κt1 1e m2 1e 2 m2 2e { m2 1m2 2ω= λ ⎡⎤⎢ + + + λ − − −2 }⎣ β⎥⎦ β−βt 2 2{ 1 t121 te ⎡ κmeω−κm2e⎤ κ}= λ ⎢ + + +⎣2 β 2 ⎥⎦Var[ JP( ζ)] = E ⎡JP ( ζ) ⎤ ( E ⎡JP( ζ) ⎤⎢ ⎥ −⎣ ⎦)⎣ ⎦2 212β−β { t 1 t 2 121 2 21 2 t2( mt mte ⎡ κ ω−κκme m e ⎤ λ e e ) }= λ ⎢ + + + − −⎣2 β 2 ⎥⎦2β2β2κ2β⎡ mtmt ⎤ ⎡ ⎤1 1 2 2κ P= ⎢ + ⎥ + ⎢⎣ ⎥⎣⎦ ⎦Var[ X(,)] ζ t Var{ m A (,0) ζ e1m A (,0) ζ e2} VarαJ () ζ2 2mt1 2 2mtm 21e Var ⎡A1( ζ,0) ⎤ m2 e Var ⎡A2( ζ,0) ⎤ α⎣ ⎦+⎣ ⎦ Var ⎡J( )2 Pζ ⎤⎣ ⎦κ2 2mt1 2 2mt= m2⎤1e Var ⎡A1( ζ, 0) ⎤ m2 e Var ⎡⎣ ⎦+⎣A2( ζ, 0)⎦+2α+ λ2κ−βt 2 21 t 1 t1 2 2{ e ⎡ κ ω−κme12 m2e ⎤ κ λ ( e mt em t+ + + − − ) }⎢⎣2 β 2 2β2 −( β−κ) t2 − ( β+κ)t= m ⎤1e Var ⎡A1( ζ, 0) ⎤ m2 e Var ⎡⎣ ⎦+⎣A2( ζ, 0)⎦+1 ω 1−{ ( ) 2 ( ) }−βt κt κtα2 λ β κ 2β κ2κ 4 β 4+ e − − e + − + e +− ⎡ − +⎣2 ( β κ) t βt ( β κ)t2αλ − − − − +e 2e e2κ ⎢⎥⎦⎤⎥.⎦22λ α2β21.4 The Time-Rate Process Equilibrium StateAt Equilibrium, t = infinite hyper-real Θ,92

Gauge Institute Journal,H. Vic DannonProof:EXζ [ ( , Θ)] ≈ 0,provided β − 4ω> 0.Var[ X α( ζ, Θ)] ≈ λ > 0 ,2 β2 2α m{ } Θ{ −α}EX [ (, ζ Θ )] = mEA [ (,0)] ζ + λ e + mEA [ (,0)] ζ λ e1 1 κ2 22m Θ1 2κ{ mEA [ ( ,0)]α}1 12 21( β β 4 ω )2= ζ + λ e− − − Θκ+{ mEA [ ( ,0)]α}2 12 2+ ζ − λ e− + − Θ.κ1( β β 4 ω )2Since we assumeβ2 2> 4ω, then,2 2β − β − 4ω> 0, and2 2β + β − 4ω> 0, andee2 2−1( β− β − 4 ω ) Θ2 ≈ 02 2−1( β+ β − 4 ω ) Θ2 ≈ 0Hence,EXζ [ ( , Θ)] ≈ 0.Var[ X( ζ, Θ )] =⎡ ⎤ +2 −( β−κ)Θm1e Var A1( ζ, 0) ⎣ ⎦≈0, since β− κ>02 − ( β+ κ)Θ+ m2 e Var ⎡A2( ζ, 0) ⎤⎣ ⎦+ ≈ 0, since β+ κ>01 −( β−κ) Θ ω −βΘ 1− ( β+ κ)Θ{ ( ) e 2 e ( ) e }2 2α+ λ − β − κ + − β + κ +2κ 4β4≈093

Gauge Institute Journal,H. Vic Dannon+ λ2α2βα 22κ λ2 ⎡ −e ( β − κ) Θ −2e β Θ −e( β + κ)Θ− − + ⎤⎢⎣⎥⎦α 2≈ λ ,2βsince β > 0.94

Gauge Institute Journal,H. Vic Dannon22.RLC Harmonic Oscillator dueto Shot Noise Voltage P(,) ζ t22.1 The Evolution Equation of RLC HarmonicProcess driven by Shot Noise Voltage P(,) ζ tCurrent fluctuations in the circuit, generate Shot NoiseVoltage P(,) ζ t that drives a random currentP (,) dqP (,) ζ ti ζ t = through the Circuit.dti (,) ζ t R = q (,) ζ t R is the Random Voltage on R .PPdiP (,) ζ tLdtis the Random Voltage on L .qP (,) ζ tCis the Random Voltage on C .95

Gauge Institute Journal,H. Vic DannonThe balance of voltages due to the shot noise isLq t + Rq t + q T = Pζ t ,1P(,) ζP(,) ζP(, ζ ) (,)Cq 1 R(,) (,) (,)1Pζ t =P(,) P ζ t −L q ζ t −L q tLC P ζ .α β ω222.2 The Shot Noise Charge q (,) ζ tR R 2 C 2−4 LC R 2 C 2−4tt−LC2L 2LC 2LCq (,) ζ t = e { A(,0) ζ e + A (,0) ζ e } +P−P1 2tτ= t ⎧ R RC 2 2−4LC ⎫ R 2 2( t τ) ⎧ RC −4LC⎫−⎪ ⎨ − ⎪⎬ − − ⎪⎨ + ⎪⎬( t−τ)2L2LC2L2LC⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭C+ { ee}2 2∫ −dP ( ζτ , )RC − 4LCτ=0Proof: By 20.2, the Harmonic Oscillator evolution equationdX 2(,) ζ t = α dP (,) ζ t −β X (,) ζ t dt −ω X (,) ζ t d t,Pis solved by the ProcessX (,) ζ t = e { A(,0) ζ e + A (,0) ζ e } +P− 1βt1κt−1κt2 2 21 2τ=tPα+ −κm1( t−τ) m2( t−τ)∫ { e e } dP( ζτ , ),τ=0Pprovided2 2κ = β − 4ω> 0,m =− β + κ,1 11 2 2m =− β − κ.1 12 2 296

Gauge Institute Journal,H. Vic DannonSubstitutingthen,1α = ,LRL22Rβ = ,L2 1ω = ,LC1 1 2 24 RC 4LC LCκ = − = − LC,mm2 2−4R R C L1 2L2LC=− + C ,2 2−4R R C L1 2L2LC=− − C ,− ⎧⎪qP( ζ, t) = e ⎨A1( ζ,0) e + A2( ζ,0)e⎪⎩R R 2 C 2−4 LC R 2 C 2−4tLC2L ⎪t−2LC 2LCt⎫⎪⎬+⎪⎭τ= t ⎧ R RC 2 2−4LC ⎫ R 2 2( t τ) ⎧ RC −4LC⎫−⎪ ⎨ − ⎪⎬ − − ⎪⎨ + ⎪⎬( t−τ)2L2LC2L2LC⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭C+ { e e } P( , )2 2∫−d ζτ.RC − 4LCτ=022.3 The Poisson Distribution of q (,) ζ t− ⎧⎪qP( ζ, t) = e ⎨A1( ζ,0) e + A2( ζ,0)e⎪⎩R R 2 C 2−4 LC R 2 C 2−4tLC2L ⎪t−2LC 2LCPt⎫⎪⎬+⎪⎭τ= t ⎧ R RC 2 2−4LC ⎫ R 2 2( t τ) ⎧ RC −4LC⎫−⎪ ⎨ − ⎪⎬ − − ⎪⎨ + ⎪⎬( t−τ)2L2LC2L2LC⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭C+ { ee}2 2∫ −dP ( ζτ , )RC − 4LCτ=0is Poisson Distributed withMeanR 2 2 4 2 2tR C −t−R C −42L 2LC 2LEq [ ( ζ, t)] = e { EA [ ( ζ,0)] e + EA [ ( ζ, 0)] eC} +P−1 2t97

Gauge Institute Journal,H. Vic Dannon⎛ RC 2 2 4LC RC 2 2 4LC2 R4t−t−−t ⎞LC −2LC2LC2L e e+ λ e2 2 + λC2 2 2 2R C −4LC ⎜ − RC + R C − 4LC RC + R C −4LC+⎜⎝⎠⎟and VarianceR 2 2 4 2 2tR C − LCt−R C −4LCLLCVar[ q ( ζ, t)] = e {Var[ A ( ζ, 0)] e + Var[ A ( ζ, 0)] eLC}P−1 2tR R2C2−4LC R2C2−4LC2 − t ⎧t− t2DC LCeL eLC1 eLC+ ⎪⎨+ −2 2RC − 4LC 2 2 2 2RC R C 4LC RC⎪− + − RC + R C − 4LC⎩⎫⎪⎬⎪⎭+2CRCD R 2RC− 2L.− 4LProof: By 21.1,mt1 2Eq [ (,)] ζ t = EA [ (,0)] ζ e + EA [ (,0)] ζ ePmt1 2mt1mtα⎛ee2 ⎞ α+ λλκ ⎜− +⎜⎝ m m ⎠⎟ω1 22−R R2C2−4LC R2C2−4LCtt−2L2LC= e { E[ A(,0)] ζ e + E[ A (,0)] ζ e2LC}1 2t⎛ R2C 2 4LC R2C 2 4LC2 R4t−t−−t ⎞LC −2LC2LC2L e e+ λ e2 2 + λC2 2 2 2RC −4LC ⎜ − RC+ RC − 4LC RC+ RC −4LC+⎜⎝⎠⎟.−( ) t( ) tVar[ qP( , t)] e β − κ −Var A1( , 0) e β +ζ = ⎡ ζ ⎤ + κ Var ⎡A2( ζ, 0) ⎤⎣ ⎦ ⎣ ⎦+2κtκtα βt e 1 eλ e⎧ −− ⎫+ ⎪⎨− + − ⎪⎬+2κ⎪⎩β − κ β β + κ⎪⎭98

Gauge Institute Journal,H. Vic Dannon2 2α β − 2ω+ λ2 βω β − 4 ω22 2 22 mt1mt22 α ⎡ee−λ κ− +⎢⎣m mκ2 21 2 ω⎤⎥⎦2−R R 2 C 2−4 LC R 2 C 2−4tt−LCL LC L= e {Var[ A( ζ, 0)] e + Var[ A ( ζ, 0)] eC} +1 2t2 2 4 2 2 42 −Rt⎧R C − LCt−R C − LCt4λCLCe L eLC1 eLC+ ⎪⎨+ −2 2RC − 4LC 2 2 2 2RC R C 4LC RC⎪− + − RC + R C − 4LC⎩⎫⎪⎬⎪⎭C R C − 2L+ λ2R 2RC−4L2⎡R R2C2−4LC R R2C2−4LC2 4 2 − t t − t − t⎤2 2λ 4C L 2e LeLC2e LeLCR C − 4LC− − +2 2RC − 4LC 2 2 2 2− RC + R C − 4LC RC + R C − 4LC2LC⎢⎣⎥⎦2.22.4 The Shot Noise Charge Steady StateAt the Steady State, t = infinite hyper-real Θ,Eq [ ( ζ, Θ)]≈ λC.P2C R C 2L3Var[ qP( ζ , )] λ ⎧L2R 2RC 4Lλ⎫⎪ −Θ ≈ ⎨− C ⎪⎬.⎪⎩−⎪⎭α LProof: By 21.2, Eq [ ( ζ, Θ)] ≈ λ = λ = λC.2Pω11LC99

Gauge Institute Journal,H. Vic Dannon2 2 2α ⎧1 β − 2ω1 ⎫Var[ q P( ζ, Θ)]≈ λ ⎨⎪−λ⎬⎪2 2β2 2ω 4⎪⎩β − 4ω ω⎪⎭21 ⎧ R21 ⎫2 2L1 − 1L LC= λ ⎪⎨−λ⎪1 R 2⎬1 12 R4LC −L 22 2⎪⎩L LC LC⎪⎭2λCRC−2L= −λLC2R 2RC−4L2 3. 22.5 Shot Noise Energy of qP (,) ζ t at the Steady State21 2 λ RC−2L1 2[ ( , )]2CPζλ222 2Eq Θ ≈ − LC +1C4R 2RC−4Lλ ,where k is Boltzmann Constant,and T is the absolute Temperature.Proof:1 2 12Eq2 Pζ Θ = qC2CPζ Θ + Eq Θ P ζ2λCRC−2LλC−2 3λ LCR RC L[ (, )] {Var[ (, )] ( [ (, )])}2 2−42λ RC−2L= − +4R 2RC−4L1 2 2 1 2λ LC λ2 2C .22.6 The Shot Noise Current i (,) ζ tPR2 22 2R C − 4 LCt2LR R C −4LCtζ2LC +−Pζ = − +2L 2LCq(,) t e ( ) A (,0) e1100

Gauge Institute Journal,H. Vic Dannon−R2 22 242 ( 4) ( ,0)R C − LCttL R R C − LC−ζ2LC +− e + A e2L2LC2τ=t2 22 2R R C −[ 4 LC2CR R C −4LC− − ]( t−τ)− ( − ) e2L2LCdP(ζτ2 2RC −4LC2L2LC∫ , )+τ=0τ=t2 22 2[R R C − 4 LC2CR R C −4LC− + ]( t−τ)− ( + ) e2L2LCd ζτ2 2RC −4LC2L2LC∫ ( , )τ=0P .Proof: By 20.3,κt1 1 2 21 12 2q(,) ζ t = e { m A(,0) ζ e + m A (,0) ζ e }P− β tτ=tα+ −κ−κ1t 2m1( t−τ) m2( t−τ)∫ { me1me2} dP( ζτ , ),τ=0−R2 22 242 ( 4) ( ,0)R C − LCttL R R C − LCζ2LC += e − + A e2L2LC1−R2 22 242 ( 4) ( ,0)R C − LCttL R R C − LC−ζ2LC +− e + A e2L2LC2τ=t2 22 2R R C −[ 4 LC2CR R C −4LC− − ]( t−τ)− ( − ) e2L2LCdP(ζτ2 2RC −4LC2L2LC∫ , )+τ=0τ=t2 22 2[R R C − 4 LC2CR R C −4LC− + ]( t−τ)+ ( + ) e2L2LCd ζτ2 2RC −4LC2L2LC∫ P ( , )τ=022.7 The Poisson Distribution of i (,) ζ tP101

Gauge Institute Journal,H. Vic Dannoni (,) ζ t = q (,) ζ tPPis Poisson Distributed withMeanR2 22 2R C − 4 LCt2LR R C −4LCtζ2LC +−Pζ = − +2L 2LCEi [ ( , t )] e ( ) EA [ ( ,0)] e1−and VarianceR2 22 242 ( 4) [ ( ,0)]R C − LCttL R R C − LC−ζ2LC ,− e + E A e2L2LCR R C −4LCPζ = − +2L 2LC2R R{ 2 C 2 − 4 LC−L LC }Var[ 2 22i ( , t )] ( ) e Var ⎡⎣A1( ζ , 0) ⎤⎦++ 2−1 14 2RC−4LR R C 4LC2L2LC−R R{ 2 C 2 − 4 LCL LC }− +2 2− 2+ ( + ) eVar⎡A2( ζ,0)⎤⎣ ⎦λ1 λ − t L 14 Re2RC−L1 14 2RC−4L+ λ12LR2C2 2RC −4LCλ2 2− RC + R C −4LCLR2 2RC + R C −4LCL2ee−R R{ 2 C 2 − 4 LC−L LC }R R{ 2 C 2 − 4 LCL LC }− +R R 2 C 2−4 LC R R R 2 C 2−4t tLCL LC L L LC−{ − } − − { +− λ ( e − 2e + etttt} t)Proof: By 21.3,mtmt1 1 2 2 (,) ζ = (,) ζ1+ (,) ζ2+q t A t m e A t m eP102

Gauge Institute Journal,H. Vic Dannonτ=tα+ −κm1( t−τ) m2( t−τ)∫ { me1me2} dP( ζτ , )τ=0is Poisson Distributed withMean⎧mtEX [ ( , t)] mEA1[1( ,0)] α ⎫ ⎧αζ = ⎪⎨ ζ + λ⎪ ⎬e + ⎪⎨mEA 2[2( ζ,0)]− λ ⎫ ⎪ ⎬⎪⎭ e⎪⎩ κ⎪⎭ ⎪⎩κ−mt1 2{ ( 2 2R R C −4LC ) [ 21( ,0)] C2 2 }R RC 2 2−t4 LC2L2LC= e − + E A ζ + λ e−2L 2LC RC −4LC{ ( 2 2R R C −4LC ) [ 22( ,0)] C2 2 }R 2 2t−RC − 4 LC2L2LC− e + E A ζ + λ e2L 2LC RC −4LCand VarianceVar[ ( , )] 2 −( β−κ)qtPζ t = m1 e Var ⎡A1( ζ , 0) ⎤⎣ ⎦+2 ( β κ)t+ m2 e− + Var ⎡ ⎣A2( ζ, 0) ⎤ ⎦+1 ω 1−{ ( ) 2 ( ) }−βt κt κt4 β 4α2 e e 2α+ − − + − + e + λ22κ λ β κ β κ− ⎡ − +⎣2 ( β κ) β ( β κ)t2αλ e − − t t2 e − e − +2κ ⎢R R C 4LC2L2LC−R R{ 2 C 2 − 4 LC−L LC }2 2− 2= ( − + ) eVar⎡A1( ζ,0)⎤⎣ ⎦+R R C 4LC2L2LCR R{ 2 C 2 − 4 LCL LC }− +( 2 2− ) 2+ + eVar ⎡A2( ζ,0)⎤⎣ ⎦++1 14 2RC−4Lλ2 2− RC + R C −4LCLe−tt⎤⎥,⎦R R{ 2 C 2 − 4 LC−L LC }tt2βt103

Gauge Institute Journal,H. Vic Dannon+ 2−1 λ − t L 14 Re2RC−L1 14 2RC−4L+ λ12LR2C2 2RC −4LCλR2 2RC + R C −4LCL2eR R{ 2 C 2 − 4 LCL LC }− +R R 2 C 2−4 LC R R R 2 C 2−4t tLCL LC L L LC−{ − } − − { +− λ ( e − 2e + et} t)22.8 The Shot Noise Current Steady StateAt Equilibrium, t = infinite hyper-real Θ,Ei [P( ζ, Θ)] ≈ 0,Proof: By 21.4,Var[ ( ζ, Θ)]≈i P1λ2RLAt Equilibrium, t = infinite hyper-real Θ,Eq [ ( ζ, Θ)] ≈ 0,P21αVar[ q P( ζ, Θ)]≈ λ = λ .2β2RL22.9 Shot Noise Energy of iP (,) ζ t at the Steady State122λEi [P( ζ, Θ)]L≈ ,4R104

Gauge Institute Journal,H. Vic Dannonwhere k is Boltzmann Constant,and T is the absolute Temperature.Proof:1 2 122 Pζ2 PζPζ λ≈0≈2RLEi [ (, Θ )] L= {Var[ i (, Θ )] + ( Ei [ (, Θ)])}L≈λ.4R22.10 RLC Harmonic Oscillator Shot Noise Energy2λ RC−3LR RC−L2 241 2 2 1 2λ LC λ2 2− +CProof :2λλ RC−2L1 2 2 1 2+ − λ LC + λ C22 24R 4R RC−4Lcurrent Shot Noise Energy2Charge Shot Noise Energyλ RC−3L= − +2R 2RC−4L1 2 2 1 2λ LC λ2 2C=.105

Gauge Institute Journal,H. Vic DannonReferences[Chandrasekhar] S. Chandrasekhar, “Stochastic Problems in Physicsand Astronomy” Reviews of Modern Physics, Volume 15, Number1,January 1943.Reprinted in “Selected Papers on Noise and Stochastic Processes” editedby Nelson Wax, Dover, 1954[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of allInfinities, and the Continuum Hypothesis” in Gauge Institute JournalVol.6 No 2, May 2010;[Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute JournalVol.6 No 4, November 2010;[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge InstituteJournal Vol.7 No 4, November 2011;[Dan4] Dannon, H. Vic, “The Delta Function” in Gauge InstituteJournal Vol.8 No 1, February 2012;[Dan5] Dannon, H. Vic, “Infinitesimal Calculus of Random Walk andPoisson Processes” posted to www.gauge-institute.org[Dan6] Dannon, H. Vic, “Einstein’s Diffusion and Probability-WaveEquations” posted to www.gauge-institute.org[Dan7] Dannon, H. Vic, “Ito’s Integral” posted to www.gaugeinstitute.org[Dan8]Dannon, H. Vic, “Evolution Equations of Random Walk andPoisson Processes” posted to www.gauge-institute.org106

Gauge Institute Journal,H. Vic Dannon[Gard] Thomas Gard, “Introduction to Stochastic DifferentialEquations” Dekker, 1988[Gardiner] Crispin Gardiner “Stochastic Methods” fourth Edition,Springer, 2009.[Gnedenko] B. V. Gnedenko, “The Theory of Probability”, SecondEdition, Chelsea, 1963.[Grimmett/Welsh] Geoffrey Grimmett and Dominic Welsh,“Probability, an introduction”, Oxford, 1986.[Hoel/Port/Stone] Paul Hoel, Sidney Port, Charles Stone, “Introductionto Stochastic Processes” Houghton Mifflin, 1972.[Hsu]Hwei Hsu, “Probability, Random Variables, & RandomProcesses”, Schaum’s Outlines, McGraw-Hill, 1997.[Inc] E. L. Inc, “Integration of Ordinary Differential equations” SeventhEdition, Oliver and Boyd, 1956.[Inc] E. L. Inc, “Ordinary Differential equations”, Longman’s, Green,1927.[Ito] Kiyosi Ito, “On Stochastic Differential Equations” Memoires ofthe American Mathematical Society, No 4. American MathematicalSociety, 1951[Karlin/Taylor] Howard Taylor, Samuel Karlin, “An Introduction toStochastic Modeling”, Academic Press, 1984.[Kuo] Hui Hsiung Kuo, “Introduction to Stochastic Integration”,Springer, 2006.[Larson/Shubert] Harold Larson, Bruno Shubert, “Probabilistic Modelsin Engineering Sciences, Volume II, Random Noise, Signals, and107

Gauge Institute Journal,H. Vic DannonDynamic Systems”, Wiley, 1979.[Lemons] Don Lemons, “An Introduction to Stochastic Processes inPhysics”, John Hopkins, 2002.[Middleton] David Middleton, “An Introduction to StatisticalCommunication Theory” McGraw-Hill, 1960.[Oksendal] Bernt Oksendal, “Stochastic Differential Equations, AnIntroduction with Applications”, Fourth Corrected Printing of the SixthEdition, Springer, 2007.[Rosenhouse] Jason Rosenhouse, “the monty hall problem” , Oxford,2009.[Stratonovich] R. L. Stratonovich, “Conditional Markov Processes andtheir Applications to the Theory of Optimal Control” Elsevier, 1968[Zwillinger] Daniel Zwillinger, “Handbook of Differential Equations”Third edition, Academic Press, 1997.http://en.wikipedia.org/wiki/Shot_noisehttp://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noisehttp://en.wikipedia.org/wiki/Flicker_noisehttp://en.wikipedia.org/wiki/White_noisehttp://en.wikipedia.org/wiki/Brownian_noisehttp://en.wikipedia.org/wiki/Stochastic_differential_equationshttp://en.wikipedia.org/wiki/Wiener_processhttp://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process108