2 Lyapunov Exponent exponent number. Lyapunov spectrum for colour conductivity of 8 WCA disks.

Transcription

1 3 2 Lyapunov Exponent exponent number Lyapunov spectrum for colour conductivity of 8 WCA disks.

2 2 3 Lyapunov exponent exponent number Lyapunov spectrum for shear flow of 8 WCA disks.

3 Using the Conjugate Pairing Rule, the self diffusion coefficient can be calculated from the maximal Lyapunov exponents in a NEMD Colour Conductivity simulation, 3 D kt N = ( ) ( λ1 + λ6 ) 2 F 3 2 c (43) In order to calculate λ min, normally an extraordinarily difficult task, we calculate the largest Lyapunov exponent for the time reversed anti-steady state.

4 t 1/2 Steady State Antisteady State reverse time -Pxy. -.5 t M Pxy(.5,f) Pxy(.5,r) time 2 3 4

5 5 η η γ 1/ The figure above compares the shear viscosity computed directly using NEMD with the value obtained using the Conjugate Pairing Rule.

6 Second Law violations in Nonequilibrium Steady States 6 For reversible deterministic N-particle thermostatted systems, we examine the question of why it is so difficult to find time reversed trajectories, that will at long times, under the application of an external dissipative field, lead to Second Law violating nonequilibrium steady states. In a nonequilibrium steady state: exp[ λ τ] µ i = j n λ ni exp[ λ τ] m λ ni > mj > mj (44) where {λ ni ;n=1,..6n} is the set of local Lyapunov exponents, for segment, i. And the ratio of the limiting (τ ) probabilities that the system is on a segment i and its conjugate anti segment, i *, is,

7 µ i µ * i = exp[ λ * τ] n λ exp[ λ τ] m λ ni* mi ni > mi > = exp[ λ τ] n λ ni exp[ λ τ] m λ ni > mi mi < 7 where we used that, = exp[ τ λ ] = exp[ 3N < α> τ] n ni τi (45) 3 6N N < α> τi= λni i= 1. (46)

8 τ =.1, γ =.5 entropy production negative p(p xyτ ) entropy production positive P xyτ =<P xy > τ We show the probability distribution of <P xy > τ. The distribution is approximately Gaussian. As can be seen the right hand tail of the distribution where <P xy > τ > consists of K-states which for a time, τ, defy the Second Law of thermodynamics.

9 9 <α> τ,pxyτ, Π(P xyτ ) <α> τ,pxyτ τ = 1.6, γ =.1.5 Π(P xyτ ) P xyτ We plot Π = ln[p(<p xy > τ ) / p(<-p xy > τ )] / 2Nτ and <α> τ,pxy, for τ=1.6 and γ =.1. These two functions are essentially linear in <P xy > τ with slopes that are very nearly

10 identical. The straight line shows a weighted least squares fit to Π(<P xy > τ ) slope(γ =.1) γ =.1 γ = slope(γ =.5) τ We graph the slope, {ln[p<p xy > τ / p<-p xy > τ ]/2Nτ}/ <P xy > τ, as a function of τ for γ=.1,.5. The corresponding results for <α> τ,pxy, are not shown here since they are independent of the averaging time τ. In determining the slopes a weighted least

11 squares fit of the data was used. We see that as τ, the slope approaches the τ- independent, slope of <α> τ,pxy as a function of <P xy > τ, which is shown by the arrow. For transient states which evolve from equilibrium at t= towards the steady state we define: τ <P xy > τ,(i), 1 τ P xy( Γ () i ( s )) ds, (47) For every such transient segment, we define the i (K) segment for which < P xy > i K τ,( ) <P xy > τ,(i). This is the Kawasaki mapped segment. where, M K Γ = M K (x,y,z,p x,p y,p z,γ) = (x, y,z, p x,p y, p z,γ) Γ (K). One can show, 11 ( ) = P ( t, Γ, γ) = exp[ il( Γ, γ) t] P ( Γ) = P ( t, Γ, γ) xy xy xy ( K) (48)

12 (4) (5) P xy. (1) Γ (5) =M (K) Γ (2) (6) (2) (3) t V2 = V1( τ) = V1( )exp[ 3Nα( s; Γ() 1 ) ds] τ (49) 2τ 1 () 1 V 3 =V 1 (2τ) = V ( )exp[ 3Nα( s; Γ ) ds]. (5) So the ratio of observing transient segments and their conjugates is:

13 µ 1* /µ 1 = V 4 /V 1 () = V 1 (2τ)/V 1 () = exp[ 3Nα( s; Γ () 1) ds], τ. (51) 2τ Logarithmic probability ratio of segments:antisegments A(τ) = τ α(s)ds 1.5 Π(τ) = p[a(τ]/p[-a(τ)] lnπ(2τ) measured slope = 95±2 ~ theoretical slope = 2N-3 = time integrated entropy production per deg of freedom = A(2τ)

14 Lagrangian form of the Kawasaki Distribution 14 Clearly one can write, exp( il( Γ) t) f( Γ, ) = f( Γ, t) (52) However, since this equation is true for all Γ it must also be true for Γ(-t), so that, exp( il( Γ( t)) t) f( Γ( t), ) = f( Γ( t), t) (53) Using a Dyson decomposition of the distribution function propagator, one can show that, t exp( il( Γ) t) = exp[ 3 Nα( Γ( s)) ds]exp[ il( Γ) t] Substituting equation (54) into (53) gives, (54)

15 t f( Γ( t), t) = exp[ 3Nα( Γ( s t)) ds]exp[ il( Γ( t)) t] f( Γ( t), ) 15 t = exp[ 3Nα( Γ( s t)) ds] f( Γ( ), ) and therefore, = t exp[ 3Nα( Γ( s)) ds] f( Γ( ), ) (55) t f( Γ( t), t) = exp[ 3 Nα( Γ( s)) ds] f( Γ( ), ) We call this equation the Lagrangian form of the Kawasaki distribution. (56)

16 16 Using the Lagrangian form of the Kawasaki distribution function. Since Γ 2 =Γ 1 (t), Γ 5 =Γ 4 (t), µ * i f( Γ1( ), ) = µ f( Γ ( ), ) i 4 1 = 2t exp 3N dsα( ( s)) Γ 1 [ ] = exp 3N α 13, 2t (57)

17 Aside: d ln V Γ() t dt ( ) V( Γ( t) )= V Γ( ) e = 3Nα() t ( ) t ds 3Nα() s 17 f( Γ( t), t)= f Γ( ), e ( ) t ds 3Nα() s VΓ ( ()) V( Γ( t) )= V Γ( ) e ( ) t ds 3Nα() s

18 REFERENCES 18 Nonlinear Response Theory: D. J. Evans and G. P. Morriss, Statistical Mechanics of NonEquilibrium Liquids (Academic Press, London, 199). Proof of Conjugate Pairing Rule: D. J. Evans, E. G. D. Cohen and G. P. Morriss, Phys. Rev., A42 (199) 599. Numerical Tests of Conjugate Pairing Rule: S. Sarman, D.J. Evans, and G. P. Morriss, Phys. Rev., A45 (1992) Green Kubo Relation for Lyapunov exponents: D.J. Evans, Phys. Rev. Letts., 69 (1992) 395. Probability of Second Law violations in Nonequilibrium Steady States: D.J. Evans, E.G.D. Cohen and G.P. Morriss, Phys. Rev. Letts., 71 (1993) 241; ibid 71 (1993) Equilibrium Microstates which Generate Second Law violating Steady States: D.J. Evans and Debra J. Searles, Phys. Rev. E, 5 (1994) 1645.