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

MIXING & POLLICOTT-RUELLE RESONANCES Correlation function between observables A and B: Statistical average of a physical observable A:

DIFFUSIVE MODES: CUMULATIVE FUNCTIONS multibaker maphard-disk Lorentz gasYukawa-potential Lorentz gas

TIME EVOLUTION OF ENTROPY coarse-grained entropy: partition of phase-space region M l into cells A S t (M l |{A}) =  k B  A P t (A) ln[P t (A)/P eq (A)] + S eq with P t (A) ≈ p(  t)  Gibbs mixing property: P t (A)  P eq (A) for t  ∞ time asymptotics for t  ∞ : P t (A) = P eq (A) +   C  exp(s  t) + … Pollicott-Ruelle resonances s  and associated eigenstates fixing the coefficients C  Selection of initial conditions by a larger system including the system of interest: problem of regression. anti-diffusion ∂ t n ≈  D ∂ l 2 n diffusion ∂ t n ≈  D ∂ l 2 n eigenstates singular in unstable directions, smooth in stable directions eigenstates singular in stable directions, smooth in unstable directions Gibbs (1902)

ENTROPY PRODUCTION coarse-grained entropy: partition of phase-space region M l into cells A S t (M l |{A}) =  k B  A P t (A) ln[P t (A)/P eq (A)] + S eq with P t (A) ≈ p(  t)  time variation over time  :   S = S t (M l |{A})  S t  (M l |{A}) entropy flow:  e  S = S t  (   M l |{A})  S t  (M l |{A}) entropy production:  i  S =   S   e  S = S t (M l |{A})  S t (M l |{   A}) Direct calculation shows that  i  S ≈  k B D n  1 (grad n) 2 with the particle density: n = P t (M l ) 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)

MOLECULAR DYNAMICS SIMULATION OF DIFFUSION J. R. Dorfman, P. Gaspard, & T. Gilbert, Entropy production of diffusion in spatially periodic deterministic systems, Phys. Rev. E 66 (2002) 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 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: lattice vector:

DECOMPOSITION INTO DIFFUSIVE MODES J. R. Dorfman, P. Gaspard, & T. Gilbert, Entropy production of diffusion in spatially periodic deterministic systems, Phys. Rev. E 66 (2002) 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:

HYDRODYNAMIC MEASURE J. R. Dorfman, P. Gaspard, & T. Gilbert, Entropy production of diffusion in spatially periodic deterministic systems, Phys. Rev. E 66 (2002) invariant hydrodynamic measure: sum rules: partition expansion in powers of the wavenumber k: measure of cell A by the nonequilibrium steady state: (~ Green-Kubo formula) (no mean drift) distance:

AB INITIO DERIVATION OF ENTROPY PRODUCTION J. R. Dorfman, P. Gaspard, & T. Gilbert, Entropy production of diffusion in spatially periodic deterministic systems, Phys. Rev. E 66 (2002) entropy production: wavenumber expansion: entropy production of nonequilibrium thermodynamics

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:  i  S ≈  k B 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. ~ gaspard Singular nonequilibrium steady state:  g. (  ) = g [ x(  ) + ∫ 0  ∞ v x (  t  ) dt ] Green-Kubo formula: D = ∫ 0 ∞ eq dt Fick’s law: neq = g [ eq + ∫ 0  ∞ eq dt ] =  D g