The Fluctuation-Dissipation Relation
We show that the requirement of internal consistency imposes a fundamental relationship between two quantities: the strength of the random force that drives the fluctuations in the velocity of a particle, on the one hand; and the coefficient representing
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The fluctuation-dissipation relation We show that the requirement of internal consistency imposes a fundamental relationship between two quantities: the strength of the random force that drives the fluctuations in the velocity of a particle, on the one hand; and the coefficient representing the dissipation or friction present in the fluid, on the other.
3.1
Conditional and complete ensemble averages
What can we say about the random force η(t) that appears in the Langevin equation (2.18)? It is reasonable to assume that its average value remains zero at all times. But what is the set over which this average is to be taken? As a result of our dividing up the system into a subsystem (the tagged particle) and a heat bath (the rest of the particles), there are two distinct averaging procedures involved: • The first is a conditional average over all possible states of motion of the heat bath, i. e., of all the molecules of the fluid, but not the tagged particle. The latter is supposed to have a given initial velocity v 0 . We shall use an overhead bar to denote averages over this sub-ensemble, namely, an ensemble of subsystems specified by the common initial condition v(0) = v 0 of the tagged particle. We shall call these conditional or partial averages. • The second is a complete average, i. e., an average over all possible states of motion of all the particles, including the tagged particle. Such averages will be denoted by angular brackets, · · ·. A complete average can be achieved by starting with the corresponding conditional average, and then performing
© The Author(s) 2021 V. Balakrishnan, Elements of Nonequilibrium Statistical Mechanics, https://doi.org/10.1007/978-3-030-62233-6_3
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3.1. Conditional and complete ensemble averages
a further averaging over all possible values of v 0 , distributed according to some prescribed initial PDF pinit (v0 ). The latter depends on the preparation of the initial state. • In writing down the Langevin equation (2.18) and its formal solution in Eq. (2.19), we have included a possible time-dependent external force, as we are also interested in the effects of such an applied force (or stimulus) upon our system. Without loss of generality, we take any such force to be applied from t = 0 onward, on a system that is initially in thermal equilibrium at some temperature T . It follows that we must identify p init (v0 ) with the Maxwellian distribution 1 itself, i. e., pinit (v0 ) ≡ peq (v0 ) =
m 2πkB T
1 2
mv02 exp − . 2kB T
(3.1)
• We shall use the notation · · · eq for complete averages taken in a state of thermal equilibrium. Averages taken in a state perturbed out of thermal equilibrium by the application of a time-dependent external force will be denoted2 simply by · · · Returning to the mean value of the random force η(t), the only assumption we need to make for the present is that its average value remains zero at all times, i. e., η(t) = 0 for all t. (3.2) From Eq. (2.19), it follows immediately that v(t) = v0 e−γt +
1 m
t
dt1 e−γ(t−t1 ) Fext (t1 ),
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