hydrodynamic modes and nonequilibrium steady states - Out of
Liouville's equation is also time-reversal symmetric. Equilibrium ... special solutions of Liouville's equation: p(r,v,t) ..... thermostated systems: no Liouville theorem.
HYDRODYNAMIC MODES AND NONEQUILIBRIUM STEADY STATES Pierre GASPARD Brussels, Belgium J. R. Dorfman, College Park S. Tasaki, Tokyo T. Gilbert, Brussels
• INTRODUCTION: POLLICOTT-RUELLE RESONANCES • DIFFUSION IN SPATIALLY PERIODIC SYSTEMS • FRACTALITY OF THE RELAXATION MODES OF DIFFUSION • NONEQUILIBRIUM STEADY STATES • CONCLUSIONS
ERGODIC PROPERTIES AND BEYOND Ergodicity (Boltzmann 1871, 1884): time average = phase-space average
1 lim T →∞ T
T
∫
A(Φ t Γ0 ) dt =
0
∫ A(Γ) Ψ (Γ) dΓ = 0
A = A Ψ0
Ψ0 stationary probability density representing the equilibrium statistical ensemble Mixing (Gibbs 1902):
A(t)B(0) =
€
€
∫
A(Φ t Γ) B(Γ) Ψ0 (Γ) dΓ → A t →∞
B
lim µ(Φ−tA ∩ B ) = µ(A ) µ(B ) t →∞
Spectrum of unitary € time evolution:
ˆ pt = Uˆ t p0 = e iG t p0
Gˆ = iLˆ
Ergodicity: € density is€ unique: Gˆ Ψ = 0 The stationary probability 0 The eigenvalue z = 0 is non-degenerate.
€
Mixing: The non-degenerate eigenvalue z = 0 is the only one on the real-frequency axis. € The rest of the spectrum is continuous.
€
Statistical average of a physical observable A(Γ):
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