Nonequilibrium fluctuation-dissipation theorem of ... - LY Chen

144113-4 L. Y. Chen J. Chem. Phys. 129, 144113 2008
Wx 0 ,x 1 , ...,x N + Wx N ,x N−1 , ...,x 0
N−1
=− x n+1 − x n F n+1 − F n .
21
n=0
The assumption of microscopic reversibility means that the
right hand side of Eq. 21 is equal to zero. Now one might
be attempted to conclude that the right hand side of Eq. 21
vanishes as dt→0 because x n+1 −x n dt and F n+1 −F n
dt. But that conclusion is invalid. In fact, it does not vanish
as long as the external force Fx,t is not independent of the
system’s coordinates. In general, we have
Wx 0 ,x 1 , ...,x N + Wx N ,x N−1 , ...,x 0
N−1
=− x n+1 − x n 2 F n ,
22
n=0
where F n Fxt n ,t n /x i . The right hand side of Eq. 22
would be zero if x n+1 −x n 2 were proportional to dt 2 , but the
stochastic nature of the system makes x n+1 −x n 2 proportional
to dt instead of dt 2 . Therefore the right hand side of
Eq. 22 does not vanish in the dt→0 limit. It is equal to
twice the dissipative work instead. This means that the CFT
and JE are invalid unless the dissipative work is vanishingly
small.
Now one more question needs to be asked: What about
the case when the external force is independent of the system’s
coordinates In such a case, the right hand side of Eq.
22 does vanish. However, with such kind of external
forces, the system may not transition from state A to state B
because of the stochastic nature of its dynamics. In the experiments
of mechanically unfolding biological molecules at
constant speed, the forces acting on the system are indeed
functions of the coordinates of the terminus atoms attached
to the tips of the atomic force microscope or the optical
tweezers. CFT and JE are inapplicable to those experiments
unless the pulling speeds are slow enough that the system is
always near equilibrium.
In summary, a new FDT has been derived from the
Brownian dynamics without invoking any assumption. This
new FDT is valid in the near-equilibrium or far nonequilibrium
regime, wherever the Brownian dynamics is valid. It is
also shown that the CFT and the JE are valid only in the
near-equilibrium regime where the dissipative work is nearly
zero.
The author thanks Robert Buckley for a critical reading
of the manuscript. He acknowledges support from a NIH
grant Grant No. SC3 GM084834 and the Texas Advanced
Computing Center.
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