TIME ASYMMETRY IN NONEQUILIBRIUM STATISTICAL MECHANICS

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TIME ASYMMETRY INNONEQUILIBRIUM STATISTICAL
MECHANICS
Pierre GASPARD Brussels, Belgium J. R. Dorfman,
College Park S. Ciliberto, Lyon T. Gilbert,
Brussels N. Garnier, Lyon D. Andrieux,
Brussels S. Joubaud, Lyon A.
Petrosyan, Lyon INTRODUCTION THE BREAKING OF
TIME-REVERSAL SYMMETRY FLUCTUATION THEOREMS FOR
CURRENTS NONLINEAR RESPONSE ENTROPY
PRODUCTION TIME ASYMMETRY OF
NONEQUILIBRIUM FLUCTUATIONS CONCLUSIONS
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BREAKING OF TIME-REVERSAL SYMMETRY Q(r,v)
(r,-v)
Newtons equation of mechanics is time-reversal
symmetric
if the Hamiltonian H is
even in the momenta. Liouville equation of
statistical mechanics, ruling the time
evolution of the probability density p is also
time-reversal symmetric. The solution of an
equation may have a lower symmetry than the
equation itself
(spontaneous symmetry breaking). Typical
Newtonian trajectories T are different from
their time-reversal image Q T Q T ?
T Irreversible behavior is obtained by weighting
differently the trajectories T and their
time-reversal image Q T with a probability
measure. Spontaneous symmetry
breaking relaxation modes of an autonomous
system Explicit symmetry breaking
nonequilibrium steady state by the boundary
conditions
P. Gaspard, Physica A 369 (2006) 201-246.
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STOCHASTIC DESCRIPTION IN TERMS OF A MASTER
EQUATION
A trajectory is a solution of Hamiltons
equations of motion G(tr0,p0)
Coarse-graining cell w in the phase space
stroboscopic observation of the trajectory with
sampling time Dt
G(nDtr0,p0) in cell wn
path or history w
w0w1w2wn-1
Liouvilles equation of the Hamiltonian dynamics
-gt reduced description in terms of the
coarse-grained states w -gt master equation
for the probability to visit the state w by the
time t Pt(w)
0 steady state
rate of the transition
due to the elementary process
-gt statistical description of the equilibrium
and nonequilibrium fluctuations
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FLUCTUATION THEOREM FOR THE CURRENTS
steady state fluctuation theorem for the
currents (2004)
fluctuating currents
ex electric currents in a nanoscopic
conductor rates of chemical reactions
velocity of a linear molecular motor
rotation rate of a rotary molecular motor
affinities or thermodynamic forces
Schnakenberg network theory (Rev. Mod. Phys.
1976) cycles in the graph of the process
-gt Onsager reciprocity relations and their
generalizations to nonlinear response
thermodynamic entropy production
D. Andrieux P. Gaspard, J. Chem. Phys. 121
(2004) 6167 J. Stat. Phys. 127 (2007) 107.
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BEYOND LINEAR RESPONSE ONSAGER RECIPROCITY
RELATIONS
generating function of the currents
(Schnakenberg network theory)
fluctuation theorem for the currents
average current
linear response coefficients
linear response coefficients
(Green-Kubo formulas)
Onsager reciprocity relations
relations for nonlinear response
higher-order nonequilibrium coefficients
D. Andrieux P. Gaspard, J. Chem. Phys. 121
(2004) 6167 J. Stat. Mech. (2007) P02006.
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FLUCTUATIONS AND MICROREVERSIBILITY
Microreversibility Hamiltons equations are
time-reversal symmetric. If G(tr0,p0) is a
solution of Hamiltons equation, then
G(-tr0,-p0) is also a solution. But,
typically, G(tr0,p0) ? G(-tr0,-p0).
Coarse-graining cell w in the phase space
stroboscopic observation of the trajectory with
sampling time Dt G(nDtr0,p0) in cell wn
path or history w w0w1w2wn-1 If w
w0w1w2wn-1 is a possible path, then wR
wn-1w2w1w0 is also a possible path. But,
again, w ? wR. Statistical description
probability of a path or history
equilibrium steady state
Peq(w0w1w2wn-1) Peq(wn-1w2w1w0)
nonequilibrium steady state
Pneq(w0w1w2wn-1) ? Pneq(wn-1w2w1w0)
In a nonequilibrium steady state, w and wR have
different probability weights. Explicit
breaking of the time-reversal symmetry by the
nonequilibrium boundary conditions
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DYNAMICAL RANDOMNESS OF TIME-REVERSED PATHS
nonequilibrium steady state P (w0
w1w2 wn-1) ? P (wn-1 w2 w1 w0) If the
probability of a typical path decays as
P(w) P(w0 w1 w2
wn-1) exp( -h Dt n ) the probability of
the time-reversed path decays as P(wR)
P(wn-1 w2 w1 w0) exp( -hR Dt n )
with hR ? h entropy per unit time
dynamical randomness (temporal disorder)
h lim
n8 (-1/nDt) ?w P(w) ln P(w) time-reversed
entropy per unit time P. Gaspard, J. Stat. Phys.
117 (2004) 599
hR lim n8 (-1/nDt) ?w P(w)
ln P(wR) The time-reversed entropy per unit time
characterizes the dynamical randomness (temporal
disorder) of the time-reversed paths.
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THERMODYNAMIC ENTROPY PRODUCTION
Second law of thermodynamics entropy S
entropy flow
entropy production
Entropy production
P. Gaspard, J. Stat. Phys. 117 (2004) 599
Property hR h
(relative entropy) equality iff
P(w) P(wR) (detailed balance) which
holds at equilibrium.
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PROOF FOR CONTINUOUS-TIME JUMP PROCESSES
Pauli-type master equation
nonequilibrium steady state t-entropy per unit
time P. Gaspard X.-J. Wang, Phys.
Reports 235 (1993) 291 time-reversed
t-entropy per unit time P. Gaspard, J.
Stat. Phys. 117 (2004) 599 thermodynamic
entropy production
Luo Jiu-li, C. Van den Broeck, and G. Nicolis, Z.
Phys. B- Cond. Mat. 56 (1984) 165 J.
Schnakenberg, Rev. Mod. Phys. 48 (1976) 571
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PROOF FOR THERMOSTATED DYNAMICAL SYSTEMS
entropy per unit time time-reversed
entropy per unit time thermodynamic entropy
production d is the diameter of the
phase-space cells T. Gilbert, P. Gaspard, and
J. R. Dorfman (2007)
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INTERPRETATION
nonequilibrium steady state thermodynamic
entropy production
entropy production
dynamical randomness of time-reversed
paths hR
dynamical randomness of paths h
P. Gaspard, J. Stat. Phys. 117 (2004) 599
If the probability of a typical path decays as
the probability of the corresponding
time-reversed path decays faster as
The thermodynamic entropy production is due to a
time asymmetry in dynamical randomness.
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DRIVEN BROWNIAN MOTION
Polystyrene particle of 2 mm diameter in a 20
glycerol-water solution at temperature 298 K,
driven by an optical tweezer.
relaxation time
trap stiffness
driving force
trap velocity
Langevin equation
u gt 0
dissipated heat
u lt 0
mean dissipated heat
D. Andrieux, P. Gaspard, S. Ciliberto, N.
Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev.
Lett. 98 (2007) 150601
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PATH PROBABILITIES OF NONEQUILIBRIUM FLUCTUATIONS
comoving frame of reference
stationary probability density
path probability
ratio of probabilities for ugt0 and ult0
heat generated by dissipation
thermodynamic entropy production
D. Andrieux, P. Gaspard, S. Ciliberto, N.
Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev.
Lett. 98 (2007) 150601
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RELATIONSHIP TO DYNAMICAL RANDOMNESS
path
path probability
algorithm of time series analysis by Grassberger
Procaccia (1980s)
(e,t)-entropy
time-reversed (e,t)-entropy
(e,t)-entropy per unit time
time-reversed (e,t)-entropy per unit time
thermodynamic entropy production
D. Andrieux, P. Gaspard, S. Ciliberto, N.
Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev.
Lett. 98 (2007) 150601
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DRIVEN BROWNIAN MOTION
sampling frequency 8192 Hz
time series
resolution
(e,t)-entropy
thermodynamic entropy production
time-reversed (e,t)-entropy
D. Andrieux, P. Gaspard, S. Ciliberto, N.
Garnier, S. Joubaud, and A.
Petrosyan, Phys. Rev. Lett. 98 (2007) 150601
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DRIVEN BROWNIAN PARTICLE
potentiel velocity
resolution
ratio of probabilities for ugt0 and ult0
dissipated heat along the random path zt
D. Andrieux, P. Gaspard, S. Ciliberto, N.
Garnier, S. Joubaud, and A. Petrosyan, Phys.
Rev. Lett. 98 (2007) 150601
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DRIVEN ELECTRIC CIRCUIT
RC circuit
thermodynamic entropy production
Joule law
D. Andrieux, P. Gaspard, S. Ciliberto, N.
Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev.
Lett. 98 (2007) 150601
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CONCLUSIONS
Breaking of time-reversal symmetry in the
statistical description
Nonequilibrium transients escape-rate
formalism fractal repeller
diffusion D (1990)

viscosity h (1995)
Nonequilibrium modes of diffusion relaxation
rate -sk, Pollicott-Ruelle resonance

Nonequilibrium work fluctuation theorem
systems driven by an external forcing

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CONCLUSIONS (contd)
Nonequilibrium steady states Explicit
breaking of time-reversal symmetry by the
nonequilibrium conditions.
Fluctuation theorem for the currents

Entropy production and temporal disorder
thermodynamic entropy production temporal
disorder of time-reversed paths hR
- temporal disorder of paths h time
asymmetry in dynamical randomness

Theorem of nonequilibrium temporal ordering as a
corollary of the second law In nonequilibrium
steady states, the typical paths are more ordered
in time than the corresponding time-reversed
paths. Boltzmanns interpretation of the second
law Out of equilibrium, the spatial disorder
increases in time.
Toward a statistical thermodynamics for
out-of-equilibrium nanosystems
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