66
L. Rondoni
piles; laser-cooled atoms; particles inside living cancer cells; particles passively
advected by dynamical membranes; in the bulk-mediated diffusion on lipid bilayers; transport of heat in nantubes, etc. Levy flights and walks are among the most
common stochastic models of such systems; nonlinear 1D oscillators chains and
polygonal billiards are among the deterministic models; exactly solvable models,
meant to understand the basic mechanisms, are also available [41].
1.9.1 Transient and Steady State Fluctuation Relations
Arguably, the major results of nonequilbrium statistical physics of the past three
decades concern an extensions of the equilibrium fluctuation theory, that originated
from a paper by Evans, Cohen and Morriss dated 1993 [42]. That paper considered
a TRI, but dissipative model of shearing fluids, known as SLLOD, which is a special
case of the following models:
˙
q i =
p i
m
+ C i F e ; ˙
p i = F i + D i F e − αp i , i = 1, . . . , N
(1.234)
Then, they proposed and tested the Fluctuation Relation (FR) that was later recongnized as a generalization of Green-Kubo relations. Informally, this first FR can be
written as:
Prob.
τ ≈ −A
Prob.
τ ≈ A
≈ exp [−Aτ ]
(1.235)
where τ is the average energy dissipation rate in long time intervals of duration
τ , in a steady state. Given the reversibility of the dynamics, the dissipation may
flcutuate and even take negative values; but the formula states that negative values
occur with a probability that, compared to that of positive values, is exponentially
small both in the size of the dissipation and of the observation times. This, analogously to Boltzmann’s arguments on the H theorem, explains the second law of
thermodyamics for these kinds of systems. Because σ is an extensive quantity, its
values are proprtional to the numbers of particles, hence very large in microscopic
terms, like the observation times are also very large compared to microscopic times.
The exponential of minus the product of these two quanties is thus practically zero,
which means that in macroscopic experiments the entropy production can never be
negative.
This relation, however, does not need to be related to thermodyamic phenomena.
It represents a property of a reversible and dissipative particle system in a nonequilbrium steady state. In particular, it may apply to small systems, or to observations of
short duration, or both. In that case, the fluctuations that are not observable in macroscopic systems may be observable. This is the case of nano-tech and bio-physical systems. It is also the case of macroscopic systems observed at a microscopic scale, such
as the gravitational wave detectors. First derived within the deterministic molecular
L. Rondoni
piles; laser-cooled atoms; particles inside living cancer cells; particles passively
advected by dynamical membranes; in the bulk-mediated diffusion on lipid bilayers; transport of heat in nantubes, etc. Levy flights and walks are among the most
common stochastic models of such systems; nonlinear 1D oscillators chains and
polygonal billiards are among the deterministic models; exactly solvable models,
meant to understand the basic mechanisms, are also available [41].
1.9.1 Transient and Steady State Fluctuation Relations
Arguably, the major results of nonequilbrium statistical physics of the past three
decades concern an extensions of the equilibrium fluctuation theory, that originated
from a paper by Evans, Cohen and Morriss dated 1993 [42]. That paper considered
a TRI, but dissipative model of shearing fluids, known as SLLOD, which is a special
case of the following models:
˙
q i =
p i
m
+ C i F e ; ˙
p i = F i + D i F e − αp i , i = 1, . . . , N
(1.234)
Then, they proposed and tested the Fluctuation Relation (FR) that was later recongnized as a generalization of Green-Kubo relations. Informally, this first FR can be
written as:
Prob.
τ ≈ −A
Prob.
τ ≈ A
≈ exp [−Aτ ]
(1.235)
where τ is the average energy dissipation rate in long time intervals of duration
τ , in a steady state. Given the reversibility of the dynamics, the dissipation may
flcutuate and even take negative values; but the formula states that negative values
occur with a probability that, compared to that of positive values, is exponentially
small both in the size of the dissipation and of the observation times. This, analogously to Boltzmann’s arguments on the H theorem, explains the second law of
thermodyamics for these kinds of systems. Because σ is an extensive quantity, its
values are proprtional to the numbers of particles, hence very large in microscopic
terms, like the observation times are also very large compared to microscopic times.
The exponential of minus the product of these two quanties is thus practically zero,
which means that in macroscopic experiments the entropy production can never be
negative.
This relation, however, does not need to be related to thermodyamic phenomena.
It represents a property of a reversible and dissipative particle system in a nonequilbrium steady state. In particular, it may apply to small systems, or to observations of
short duration, or both. In that case, the fluctuations that are not observable in macroscopic systems may be observable. This is the case of nano-tech and bio-physical systems. It is also the case of macroscopic systems observed at a microscopic scale, such
as the gravitational wave detectors. First derived within the deterministic molecular