AB INITIO DERIVATION OF ENTROPY PRODUCTION

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AB INITIO DERIVATION OF ENTROPY PRODUCTION
Pierre GASPARD Brussels, Belgium J. R. Dorfman,
College Park S. Tasaki, Tokyo T. Gilbert,
Brussels MIXING POLLICOTT-RUELLE
RESONANCES COARSE-GRAINED ENTROPY ENTROPY
PRODUCTION DECOMPOSITION INTO HYDRODYNAMIC
MEASURES AB INITIO DERIVATION OF ENTROPY
PRODUCTION CONCLUSIONS
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MIXING POLLICOTT-RUELLE RESONANCES
Correlation function between observables A and B
Statistical average of a physical observable A
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DIFFUSIVE MODES CUMULATIVE FUNCTIONS
multibaker map
hard-disk Lorentz gas
Yukawa-potential Lorentz gas
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TIME EVOLUTION OF ENTROPY
coarse-grained entropy partition of
phase-space region Ml into cells A
St(Ml A) - kB SA Pt(A) lnPt(A)/Peq(A)
Seq with Pt(A) p(Gt) DG Gibbs mixing
property Pt(A) Peq(A) for t 8
time asymptotics for t 8 Pt(A)
Peq(A) Sa Ca exp(sa t)
Pollicott-Ruelle resonances sa and associated
eigenstates fixing the coefficients Ca
Gibbs (1902)
eigenstates singular in unstable
directions, smooth in stable directions
eigenstates singular in stable
directions, smooth in unstable directions
anti-diffusion ?tn - D ?l2n
diffusion ?tn D ?l2n
Selection of initial conditions by a larger
system including the system of interest problem
of regression.
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ENTROPY PRODUCTION
coarse-grained entropy partition of
phase-space region Ml into cells A
St(Ml A) - kB SA Pt(A) lnPt(A)/Peq(A)
Seq with Pt(A) p(Gt) DG time
variation over time t DtS St(Ml A) -
St-t(Ml A) entropy flow
DetS St-t(F-tMl A) - St-t(Ml
A) entropy production DitS DtS
- DetS St(Ml A) - St(Ml Ft A) Direct
calculation shows that DitS t kB D n-1 (grad
n)2 with the particle density
n Pt(Ml) because of the singular character
of the nonequilibrium states J. R. Dorfman, P.
Gaspard, T. Gilbert, Entropy production of
diffusion in spatially periodic deterministic
systems, Phys. Rev. E 66 (2002) 026110
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MOLECULAR DYNAMICS SIMULATION OF DIFFUSION
Hamiltonian dynamics with periodic boundary
conditions. N particles with a tracer particle
moving on the whole lattice. The probability
distribution of the tracer particle thus
extends non-periodically over the whole lattice.
lattice vector
lattice Fourier transform
first Brillouin zone of the lattice
initial probability density close to equilibrium
time evolution of the probability density
lattice distance travelled by the tracer
particle
J. R. Dorfman, P. Gaspard, T. Gilbert, Entropy
production of diffusion in spatially periodic
deterministic systems, Phys. Rev. E 66 (2002)
026110
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DECOMPOSITION INTO DIFFUSIVE MODES
measure of a cell A at time t
with
dispersion relation of diffusion
hydrodynamic measure at time t
invariance under time evolution de Rham-type
equation
J. R. Dorfman, P. Gaspard, T. Gilbert, Entropy
production of diffusion in spatially periodic
deterministic systems, Phys. Rev. E 66 (2002)
026110
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HYDRODYNAMIC MEASURE
invariant hydrodynamic measure
expansion in powers of the wavenumber k
measure of cell A by the nonequilibrium steady
state
sum rules partition
distance
(no mean drift)
( Green-Kubo formula)
J. R. Dorfman, P. Gaspard, T. Gilbert, Entropy
production of diffusion in spatially periodic
deterministic systems, Phys. Rev. E 66 (2002)
026110
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AB INITIO DERIVATION OF ENTROPY PRODUCTION
entropy production
wavenumber expansion
entropy production of nonequilibrium
thermodynamics
J. R. Dorfman, P. Gaspard, T. Gilbert, Entropy
production of diffusion in spatially periodic
deterministic systems, Phys. Rev. E 66 (2002)
026110
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CONCLUSIONS
In the long-time limit, the approach to
equilibrium is controlled by the Pollicott-Ruelle
resonances (including the dispersion relation of
diffusion) and the associated eigenstates
(including the diffusive modes). The same applies
to the coarse-grained entropy. Ab initio
derivation of the entropy production expected
from nonequilibrium thermodynamics
DitS t kB D n-1 (grad n)2
(2002) because the diffusive
modes are singular and break the time-reversal
symmetry. This result is obtained in the limit of
long times and low wavenumbers, where the
diffusive mode gives the singular distribution of
the nonequilibrium steady state. This latter
appears as part of the Green-Kubo formula giving
the diffusion coefficient D.
Singular nonequilibrium steady state Yg.(G)
g x(G) ?0- 8 vx(Ft G) dt Green-Kubo
formula D ?08 ltvx(0)vx(t)gteq dt Ficks
law ltvxgtneq g ltvx xgteq ?0- 8
ltvx(0)vx(t)gteqdt - D g
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