FINITE-TIME STOCHASTIC THERMODYNAMICS AND OPTIMAL MASS TRANSPORT Krzysztof Gawedzki , Vienna , May 2012 ”Time is the longest distance between two places” Tennessee Williams Stochastic Thermodynamics : • In classical version it describes dynamics of mesoscopic systems (colloids, polymers, biomolecules, etc.) in contact with heat bath(s) modeled by random noise • Subject with long history starting with Einstein , Smoluchowski , Langevin • More recently revived in the context of theoretical study of fluctuation relations : Kurchan , Lebowitz - Spohn , Jarzynski , Crooks , Sekimoto , Maes , Seifert , . . .
• A simple set-up for studying interplay between thermodynamical and statistical concepts away from equilibrium • In quantum version it uses Markovian modelization of the dynamics of open nanoscopic systems • Lends itself to experimental verifications, e.g. in experiments by Stuttgart ( Bechinger ), Lyon ( Ciliberto ), Barcelone ( Ritort ), Berkeley ( Bustamante ), Notre Dame ( Orlov ), . . . groups The simplest classical model: overdamped Langevin equation d x = − M ∇ U ( t, x ) + η ( t ) dt with constant mobility matrix M = ( M ij ) > 0 and the white noise = 2 k B T M ij δ ( s − t ) η i ( s ) η j ( t ) � � տ Einstein relation
1 st Law of Stochastic Thermodynamics • fluctuating work performed in time interval [0 , t f ] : � t f W = ∂ t U ( t, x ( t )) dt 0 • fluctuating heat dissipation: � t f ∂ i U ( t, x ( t )) ◦ dx i ( t ) Q = − 0 ( with ” ◦ ” marking the Stratonovich convention ) W − Q = U ( t f , x ( t f )) − U (0 , x (0)) ≡ ∆ U holds trajectory-wise, not only for the means !
2 nd Law of Stochastic Thermodynamics The probability density − R ( t, x ) � � � � ρ ( t, x ) = δ ( x − x ( t )) ≡ exp k B T evolves according to the Fokker - Planck equation that may be written as the advection equation ∂ t ρ + ∇ · ( ρ v ) = 0 in the field ( Nelson 1967) current velocity δ ( x − x ( t )) ◦ d x � � dt ( t ) v ( t, x ) = = M ∇ ( R − U ) ρ ( t, x ) ( again with the Stratonovich convention )
2 nd Law of Stochastic Thermodynamics (cont’d) • The fluctuating instantaneous entropy of the system is 1 S sys ( t ) = − k B ln ρ ( t, x ( t )) = T R ( t, x ( t )) with the mean given by the Gibbs - Shannon formula � � � S sys ( t ) = − k B ρ ( t, x ) ln ρ ( t, x ) d x and the change along the trajectory � t f 1 d ∆ S sys = dt R ( t, x ( t )) dt T 0 • The change of entropy of the system is accompanied by the change of entropy of the thermal environment given by the thermodynamical relation � t f = − 1 Q ∂ i U ( t, x ( t )) ◦ dx i ( t ) ∆ S env = T T 0
2 nd Law of Stochastic Thermodynamics (cont’d) • The total change of fluctuating entropy ∆ S tot = ∆ S sys + ∆ S env satisfies the Jarzynski equality (one of Fluctuation Relations ) � e − ∆ S tot /k B � = 1 (an easy exercise based on Girsanov and Feynman - Kac formulae) implying by the Jensen inequality the 2 nd Law stating that � � ∆ S tot ≥ 0 that also follows by a direct calculation giving � t f 1 � v ( t, x ) · M − 1 v ( t, x ) ρ ( t, x ) d x � � ∆ S tot = dt T 0
Landauer Principle ( IBM Journal of Res. and Dev. 5:3 (1961) ) Erasure of one bit of memory in a computation in thermal environ- ment requires dissipation of at least k B T ln 2 of heat (in mean) Model : overdamped Langevin evolution from from the initial state Rf ( x ) − Ri ( x ) − 1 1 kBT kBT ρ i = Z i e to the final state ρ f = Z f e with ρ i (red) and ρ f (blue) R i (red) and R f (blue)
• at the initial time t = 0, x (0) is either in the left or in the right potential well (1 bit of information ) • at the final time t = t f , x ( t f ) is in the right potential well with no memory of where it started • The Landauer bound � � ≥ k B T ln 2 Q is implied by the 2 nd Law rewritten as the bound � � � � ≥ − T ∆ S sys Q since here ≈ − k B ln 1 + 2 k B (ln 1 2 ) 1 � � ∆ S sys = − k B ln 2 2
Finite-time Thermodynamics The 2 nd Law & Landauer bounds are saturated in quasi-stationary • processes that take infinite time ( if ρ i � = ρ f ) • In computation, one wants to minimize dissipated heat but also to go fast • This gives rise to the question: Given ρ i , ρ f and the length t f of the time � � window, what is the minimal ∆ S tot ? • Problems studies in the thermal engineering theory from the 50’ by Novikov , Chambadal , Curzon - Ahlborn , . . . , and, after the first oil crisis, by Berry , Salamon , Andresen . . . who coined the name of Finite-Time Thermodynamics • In the context of Stochastic Thermodynamics , they were first addressed by Schmiedl - Seifert in 2007 for Gaussian processes
Main result: Aurell - Mej` ıa-Monasterio - Muratore-Ginanneschi (2011), Aurell - G. - Mej` ıa-Monasterio - Mohayaee - Muratore-Ginanneschi (2012) but otherwise arbitray control potentials U ( t, x ), For fixed ρ i , ρ f , t f 1 � � ∆ S tot min = t f T K min where K min = min K [ x f ( · )] over maps x i �→ x f ( x i ) carring ρ i to ρ f , i.e. such that ρ i ( x i ) d x i = ρ f ( x f ) d x f , of the quadratic cost function � ( x f ( x i ) − x i ) · M − 1 ( x f ( x i ) − x i ) ρ i ( x i ) d x i K [ x f ( · )] = • Minimization of K [ x f ( · )] over the maps x i �→ x f ( x i ) that transport ρ i to ρ f is the celebrated Monge (1781) - Kantorovich (1942) Optimal Mass Transport Problem
Proof. A corollary of the result of Benamou - Brenier (1997) relating the optimal mass transport to the Burgers equation • Benamou - Brenier minimize the functional t f � � ( v · M − 1 v )( t, x ) ρ ( t, x ) d x A [ ρ, v ] = dt 0 over densities ρ ( t, x ) and velocity fields v ( t, x ) satisfying advection equation ∂ t ρ + ∇ · ( ρ v ) = 0 and such that ρ (0 , x ) = ρ i ( x ) , ρ ( t f , x ) = ρ f ( x ) • The advection equation with the above initial conditions is solved by � ρ ( t, x ) = δ ( x − x ( t ; x i )) ρ i ( x i ) d x i for the Lagrangian flow of v ( t, x ) d x dt ( t ; x i ) = v ( t, x ( t ; x i )) , x (0; x i ) = x i
• Inserting this solution to the expression for A gives: � t f � � d x dt · M − 1 d x � A [ ρ, v ] = ( t ; x i ) ρ i ( t, x i ) d x i dt dt 0 • Minimizing first over the curves [0 , t f ] ∋ t �→ x ( t ; x i ) keeping x ( t f ; x i ) = x f ( x i ) fixed, with the minima attained on straight lines t ≡ x lin ( t ; x i ) , � � x ( t ; x i ) = x i + x f ( x i ) − x i t f x f ( x i ) − x i d x dt ( t, x i ) = one reduces the minimization of A with to t f the optimal mass transport problem considered before: 1 A min = K min t f
• The map x i �→ x f ( x i ) that minimizes the quadratic cost function is of the gradient type : x f ( x i ) = M · ∇ F ( x i ) for a convex function F • minimizing A The velocity field v has the linear Lagrangian flow x lin ( t ; x i ) , and, as such, satisfies the inviscid Burgers equation ∂ t v + ( v · ∇ ) v = 0 , It is necessarily also of the gradient type ! • v ( t, x ) = M ∇ Ψ( t, x ) where, for x = x lin ( t ; x i ), 1 2 t f x i · M − 1 x i + 1 1 2 t ( x − x i ) · M − 1 ( x − x i ) − Ψ( t, x ) = t f F ( x i ) ∂ t Ψ + 1 2 ∇ Ψ · M ∇ Ψ = 0
• It follows that v = M ∇ Ψ minimizing A is the current velocity = M ∇ ( R − U ) for the overdamped Langevin process such that U ( t, x ) = R ( t, x ) − Ψ( t, x ) for � δ ( x − x lin R ( t, x ) = − k B T ln ( t ; x i )) ρ i ( x i ) d x i f • Since 1 � � ∆ S tot = T A [ ρ, v ] for v = M ∇ ( R − U ), we conclude that 1 1 � � ∆ S tot min = T A min = t f T K min even if, a priori , A was minimized without assuming the gradient form of v • The optimal protocol U ( t, x ) is given by the formulae on the top �
Geometric interpretation a la Jordan - Kinderlehrer - Otto (1998) ` • K min d W ( ρ i , ρ f ) is the square of the Wasserstein distance corresponding to the formal Riemannian metric � � ∂ t ρ � 2 ( v · M − 1 v )( x ) ρ ( x ) d x W = min v such that ∂tρ + ∇ ( ρ v )=0 on the space of densities ρ • The Fokker - Planck equation describes the gradient flow corresponding to the free energy functional � � F t [ ρ ] = U ( t, x ) ρ ( x ) d x + k B T ρ ( x ) ln ρ ( x ) d x • One has � t f 1 1 � ∂ t ρ ( t, · ) � 2 t f T d W ( ρ i , ρ f ) 2 � � ∆ S tot = W dt ≥ T 0 • Optimal protocol gives the geodesic joining ρ i to ρ f
Corollary (Finite-time refinement of the 2 nd Law ) • For overdamped Langevin process evolving in time interval [0 , t t ] with the initial probability density ρ i and the final one ρ f 1 � � ∆ S tot ≥ t f T K min ≥ 0 with the left lower bound saturated by the protocol with U = R − Ψ • Equivalently 1 � � � � ≥ − T ∆ S sys + K min Q t f � � ∆ S sys = − k B ln 2 and for we obtain a finite time refinement of the Landauer Principle
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