Statistical Mechanics - Physics at Oregon State University

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4 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS. have to be able to define an outside heat reservoir. Therefore, even though in statistical mechanics we will be able to evaluate the entropy analogue for very small systems or of the largest possible system, the universe, we cannot equate this calculated quantity with the entropy used in thermodynamics. Microscopic origin of model parameters. In thermodynamics one postulates relations between all the parameters (equations of state) and formulas for different forms of the energy (state functions). Typical equations of state are: for an ideal gas, or the van der Waals form pV = NRT (1.1) 2 N (p + a V 2 )(V − Nb) = NkBT (1.2) Based on macroscopic considerations, the parameters a and b in the second equation are simply parameters which could be fit to the experimental results. They can only be given a definite meaning in a discussion of the microscopic aspects of the models. For example, using experimental data one derives a value for b of about 1 m3 /atom. This is not a problem in thermodynamics, but it is in statistical mechanics, where the parameter b is equal to the volume of an atom. Having this statistical mechanics background available, one can exclude certain ranges of parameter values as unrealistic. Next, the equations of thermodynamics are used together with these equations of state to describe the properties of the system. Phase transitions are possible when using the van der Waals equation, but not with the ideal gas law. 1.2 Program of statistical mechanics. What do we do? Statistical mechanics aims to derive all these macroscopic relations from a microscopic point of view. An atomic model is essential in this case; thermodynamics can be derived without any reference to the atomic nature of matter. Since the number of particles is very large in many cases (we always have to take the limit N → ∞ ), one can only discuss average quantities. The program of statistical mechanics can therefore be described by the following four steps: 1. Assume an atomic model. 2. Apply the equations of motion (Newton, Schrödinger, etc). 3. Calculate average quantities.
1.3. STATES OF A SYSTEM. 5 4. Take the thermodynamic limit and compare with experiment/thermodynamics. Collective versus random behavior. For example, if one could solve all equations of motion, one would find ri(t) and pi(t) for all particles i = 1, · · · , N. It is then easy to separate collective motion and random motion: pi(t) = pav(t) + δpi(t) (1.3) Energy related to this average motion can easily be retrieved. For example, the energy stored in moving air can be used to drive a windmill. The energy stored in the random components δpi(t) of the motion is much harder to get out. Heat engines have a well defined upper limit (Carnot limit) to their efficiency! Also, the amount of data needed to describe these random components is much too large. This is where entropy and temperature play a role. In some sense, entropy measures the amount of randomness in a system. The energy stored in the random components of the motion, δpi(t) is related to thermal energy. It is hard to recover, it is hard to measure. The only viable description we have available is a statistical one. we therefore define entropy as a measure of this randomness. Entropy measures everything we do not know easily, or therefore everything we do not know. Entropy measures non-information. Entropy first! In statistical mechanics we define entropy first, and then derive temperature. This is the opposite from the situation in thermodynamics. Randomness involves the number of states available for a system, and states are much easier to count using quantum mechanics. This is the basic reason why quantum statistical mechanics is so much easier than classical statistical mechanics. In a quantum mechanical description we have to count numbers of states, in classical mechanics we have to integrate over regions of phase space, which is harder to describe exactly. Ideas similar to entropy are used in different fields like information theory where the amount of information in a message can also be related to a concept like entropy. In general, any system which contains many degrees of freedom which we cannot know and are not interested in can be described using a concept similar to entropy for these degrees of freedom! 1.3 States of a system. Ising model.
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Statistical Mechanics - Physics at Oregon State University

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