Fluctuations II p. 1/31 Fluctuation relations and nonequilibrium thermodynamics II Alberto Imparato and Luca Peliti Dipartimento di Fisica, Unità CNISM and Sezione INFN Politecnico di Torino, Torino (Italy) Dipartimento di Scienze Fisiche, Unità CNISM and Sezione INFN Università Federico II, Napoli (Italy)
Fluctuation relations for the work Fluctuations II p. 2/31
Fluctuations II p. 3/31 A Markov chain x {1, 2,...,q} t {0, 1, 2,...} Equilibrium distribution determined by hamiltonian H(x, µ): P eq (x,µ) = e βh(x,µ) Z µ Z µ = x e βh(x,µ) = e βf(µ) Evolution equation for the probabilities: p(x,t + 1) = x W xx (µ(t))p(x,t) x W xx (µ)p eq (x,µ) = P eq (x,µ)
Work and heat E(µ) = H(µ) µ A µ = x A(x)P eq (x,µ) Manipulation: µ µ + dµ: de(µ) = dh(x,µ)p eq (x,µ(t)) + H(x,µ) dp eq (x,µ) x x }{{}}{{} dw dq ds eq ( ) = k B d P eq (x,µ) ln P eq (x,µ) x = k B ln P eq (x,µ) dp eq (x,µ) x Fluctuations II p. 4/31
Fluctuations II p. 5/31 Path work and heat under manipulation Path x = (x(0),x(1),x(2),...,x(t )) Manipulation protocol: µ = (µ(0),µ(1),...,µ(t 1)) W[x] = Q[x] = T 1 t=0 T 1 t=0 [H(x(t + 1),µ(t + 1)) H(x(t + 1),µ(t))] [H(x(t + 1),µ(t)) H(x(t),µ(t))] E = H(x(T ),µ(t )) H(x(0),µ(0)) = W + Q N.B. W and Q depend on the whole path, E depends only on initial and final states
Fluctuations II p. 6/31 Path probability H 0 1 2 3 4 t P[x x(0)] = W x(t ),x(t 1) (µ(t 1)) W x(2),x(1) (1)W x(1),x(0) (µ(0)) N.B.: The transitions take place with the old probability W xx (µ(t 1)), then the energy changes to H(x, µ(t))
Fluctuations II p. 7/31 Reversed path Time-reversal invariant states: Ix = x Time-reversed path: x = (x(0),x(1),...,x(t 1),x(T )) x = (x(t ),x(t 1),...,x(1),x(0)) x(t) = x(t t) = x( t) Time-reversed protocol: µ : µ(t) = µ(t t) = µ( t) t = 1,...,T Probability of the reversed transition: Ŵ xx (µ) : Ŵ xx (µ)p eq (x,µ) = W x x(µ)p eq (x,µ)
Fluctuations II p. 8/31 Detailed balance In particular, if detailed balance holds: W xx (µ) e βh(x,µ) = W x x(µ) e βh(x,µ) one has Ŵ xx (µ) = W xx (µ)
Fluctuations II p. 9/31 Probability of the reversed path W xx (µ(t)) = W xx (t) Ŵ xx (t) = W x x(t t) etc. t = 1,...T P[ x x(0)] = Ŵ x(t ), x(t 1)(T ) Ŵ x(2), x(1)(2)ŵ x(1), x(0)(1) = W x(1),x(0) (0)e β(h(x(1),µ(0)) H(x(0),µ(0))) W x(2),x(1) (1)e β(h(x(2),µ(1)) H(x(1),µ(1))) W x(t ),x(t 1) (T 1) β(h(x(t ),µ(t 1)) H(x(T 1),µ(T 1))) e = P[x x(0)] e βq[x]
Fluctuations II p. 10/31 Unconditional probabilities: Crooks s relation P[x] = P[x x(0)]p eq (x(0),µ(0)) etc. Thus P[ x] = P[ x x(0)]e βh( x(0), µ(0))/ Z µ(0) = P[x x(0)]e β{q[x] H(x(T ),µ(t ))}/ Z µ(t ) = P[x] e β{q[x] (H(x(T ),µ(t )) H(x(0),µ(0)))} ( Z µ(0) / Zµ(T ) ) Q[x] H = W[x] Z µ(0) / Zµ(T ) = e β F P[ x] = P[x] e β(w[x] F)
Fluctuations II p. 11/31 Jarzynski s equality (JE) Summing over x: e βw = x P[x] e βw[x] = e β F x β F P[ x] = e Comments: F = F(µ(T )) F(µ(0)) is the change in the equilibrium free energy with the corresponding values of the parameter µ The system is not at equilibrium at the end of the process One assumes that the system is initially at equilibrium The average is over all realizations of the process
Fluctuations II p. 12/31 Sudden change in the hamiltonian µ 0 µ 1 W[x] = H(x(0),µ 1 ) H(x(0),µ 0 ) e βw = x = 1 Z µ0 e β(h(x,µ 1) H(x,µ 0 )) e βh(x,µ 0) x e βh(x,µ 1) = Z µ 1 Z µ0 Z µ0
Fluctuations II p. 13/31 Adiabatic change µ(t) = l(t/t ) l(0) = µ 0 l(1) = µ 1 W[x] = = = T 1 t=0 t=0 T (H(x(t + 1),µ(t + 1)) H(x(t + 1),µ(t))) 1 T 1 H(x(t + 1),µ) dl T µ µ=l(t/t ) dx H(x(t dt ),µ) µ(t ) 0 µ µ=µ(t ) H(x,µ) dµ = W th µ 0 µ µ1 µ x=t/t
Fluctuations II p. 14/31 Slow parameter change W β 1 ln e βw W d = W F 0 e βw = e β( W +δw) e β W β e δw = e 1 β W β δw + 1 2 β2 δw 2 + e β W ( 1 + β2 2 ) δw 2 + Thus W d β 2 δw 2
Fluctuations II p. 15/31 Dissipation and irreversibility e β(w[x] F) = P[x] P[ x] Thus W d = W F = β 1 P[x] ln P[x] x P[ x] }{{} D(P[x] P[ x]) van den Broeck et al., 2007
Fluctuations II p. 16/31 Work distribution Define P(W) = x P(W) = x δ(w W[x]) P[x] δ(w W[x]) P[x] Then W d = β 1 dw P(W) ln P(W) P( W) = β 1 D(P(W) P( W)) The whole mutual information of P[ x] vs. P[x] is contained in the work distribution!
Fluctuations II p. 17/31 Collective coordinate Exploring equilibrium free-energy landscapes: Collective coordinate: M(x); M eq = M eq Manipulation: U (M(x),µ) H(x,µ) = H 0 (x) U(M(x),µ) Target: Free energy of the M landscape F 0 (M) = β 1 ln dx δ(m M(x)) e βh 0(x) Probability distribution for M: P eq (M) = e β(f 0(M) F 0 )
Work probability distribution Evolution operator L µ : P(x,t) = L µ(t) P(x,t) t L µ P eq (x,µ) = 0 Work: W = t 0 dt µ(t ) U (M(x(t )),µ(t )XS) µ Φ(x,W,t): Joint probability of x and of the accumulated work W : Φ(x,W,t) t = L µ(t) Φ + µ(t) U(M(x),µ(t)) µ Φ WFluctuations II p. 18/31
Fluctuations II p. 19/31 The generating function Define Ψ(x,λ,t) = dw e λw Φ(x,W,t) Then Ψ t = L µ(t)ψ + λ µ(t) U(M(x),µ(t)) µ For λ = β one obtains: Ψ Ψ(x, β,t) = e βh(x,µ(t)) Z µ(0) (1) dx Ψ(x, β,t) = e βw = Z µ(t) Z µ(0)
Proof of (1) Define Then ψ(x, t) satisfies ψ(x,t) = e βh(x,µ(t)) Z µ(0) ψ(x, 0) = P eq (x,µ(0)) = Ψ(x, β, 0) ψ t = β µ H(x,µ(t)) ψ(x, t) µ = L µ(t) ψ(x,µ(t)) β µ H(x,µ(t)) ψ(x, t) µ since ψ(x,t) P eq (x,µ(t)) and L µ P eq (x,µ) = 0: thus ψ(x,t) = Ψ(x, β,t) Fluctuations II p. 20/31
Fluctuations II p. 21/31 The basic identity Multiply (1) by δ(m M(x)) and integrate on x: δ(m M(x))e βw = dx δ(m M(x)) e βh(x,µ(t)) Z 0 = exp { β [F 0 (M) U(M,µ(t)) F 0 ]} Multiply both sides by e βu(m,µ(t)) : e βu(m,µ(t)) δ(m M(x))e βw = e β[f 0(M) F 0 ] Crooks 1999, Hummer and Szabo 2001
Fluctuations II p. 22/31 Criticism Disproving Jarzynski s equality is a minor industry E. G. D. Cohen and D. Mauzerall, J. Stat. Mech. P07006 (2004) Jeayoung Sung, cond-mat/0506214, cond-mat/0510119, cond-mat/0512250 D. H. E. Gross, cond-mat/0508721 B. Palmieri and D. Ronis, Phys. Rev. E 75, 011133 (2007) J. M. G. Vidal and J. M. Rubi, arxiv: 0704.0761, arxiv: 0707.3802, Phys. Rev. Lett. 100, 020601 (2008)
Fluctuations II p. 23/31 An example: The free expansion V V F = T S = k B T ln 2 W = 0
Fluctuations II p. 23/31 An example: The free expansion V V F = T S = k B T ln 2 W = 0 V V µ S = 0 (!)
Fluctuations II p. 24/31 The correct way v V X Work distribution for compression (left) and expansion (right) for a fixed F and varying piston velocity v. Dashed lines are molecular dynamics simulations. Bena, van den Broeck, Kawai, cond-mat/0506289
A gas with a single molecule R. H. Lua and A. Y. Grosberg, cond-mat/0502434 β, m, τ = 1 v = 2v p v w (n) = 1 [v (n 1)2 v (n)2] = 2v p (v p v (n 1)) 2 W(v, n) = n k=1 w (n) = 2nvv p 2n 2 v 2 p Fluctuations II p. 25/31
Fluctuations II p. 26/31 Fast moving piston v p 1 L v p n = 0,1 v(w) = v p + W x > L (v v p ) = L W 2v p 2v p ( ) 2 v p + 2vp W W 2πvp P(W) δ(w) P 0 + e 1 2 e W = 1 + v p L P W>0 1 2πLv 2 p e v2 p /2 } {{ 4v p L } P W>0 W 4 2πLv 2 p e v2 p /2 = 4 P W>0
Fluctuations II p. 27/31 On the concept of work H(x,µ) = H 0 (x) µ(x x 0 ) H 0 (x) = 1 2 kx2 Thermodynamical work: W th = = µ 0 µ 0 dµ x x 0 µ ( ) µ dµ k x 0 = µ2 2k + µx 0 k H 0 = H 0 µ H 0 0 = 1 2 k ( x 2 µ x 2 0 ) = µ2 2k
Fluctuations II p. 28/31 A physical example F=qE +σ σ O x x0 ẋ = Γ[ kx + µ(t)] + η(t) ẋ µ k W qs 0 = t 0 µ(t ) ẋ(t ) = µ2 2k
Fluctuations II p. 29/31 W vs. W 0 W 0 is the work of the applied force on the system W is the work of the system on the environment If H = H 0 U(x,µ), then W 0 W = U(x(t),µ(t)) U(x(0),µ(0)) W 0 satisfies a different fluctuation relation: e βw 0 0 = 1 A[x] 0 = x A[x]P[x x(0)] e βh 0(x(0)) Z 0 Bochkov and Kuzovlev, 1977
Fluctuations II p. 30/31 Proof of Bochkov and Kuzovlev s relation Sample with the free probability P 0 : P[x x(0)] P[ (x) x(0)] = e βq[x] P 0 (x(0)) = e βh 0(x(0)) Z 0 P0 (x(0)) = e βh0(x(0)) Z 0 = P 0 (x(0)) Thus, defining P 0 = P[x x(0)]p 0 (x(0)) etc., P 0 [x] P 0 [ x] = eβ( Q[x]+ H 0(x(t))) Q + H 0 = H 0 + U + W H 0 = U + W = W 0 e βw 0[x] 0 = x P 0 [ x] = P 0 [x]e βw 0[x] P 0 [x]e βw 0[x] = x P 0 [ x] = 1
Fluctuations II p. 31/31 Bibliography C. Bustamante, J. Liphardt, F. Ritort, arxiv:cond-mat/0511629 (2005); F. Ritort, Poincaré Seminar 2, 193-226 (2003) C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997); Phys. Rev. E 56, 5018 (1997); J. Stat. Phys. 96, 77 (2000) G. E. Crooks, Phys. Rev. E 60 2721 (1999); 61, 2361 (2000); Thesis (UC Berkeley, 1999) R. C. Lua, A. Y. Grosberg, J. Phys. Chem. B 109, 6805 (2005) R. Kawai, J. M. R. Parrondo, C. van den Broeck, Phys. Rev. Lett. 98, 080602 (2007); A. Gomez-Martin, J. M. R. Parrondo, C. van den Broeck, arxiv:0710.4290 (2007) I. Bena, C. van den Broeck, R. Kawai, Europhys. Lett. 71, 879 (2005) G. N. Bochkov, Y. E. Kuzovlev, Physica A 106 443, 480 (1981)