First passage fluctuation relations ruled by cycles affinities F. Cornu ⋆ joint work with M. Bauer ⋆⋆ ⋆ Laboratoire de Physique Théorique, Orsay ⋆⋆ Institut de Physique Théorique, CEA Saclay J. Stat. Phys. (2014) 155 703
STOCHASTIC PROCESSES OF INTEREST Semi-Markovian property Bauer & Cornu : First passage FR & cycle affinities Semi-Markovian processes Firenze, 2014/05/30 2 / 30
1.1 Example of processes of interest : a bacterial ratchet motor Di Leonardo & al. PNAS, 107 9541 (2010) • Experiment : asymmetric gear (diameter : 48 µ m , thickness 10 µ m ) in active bath of self-propelling bacteriae. α t : angle of black spot position at time t � α t t � = 1 revolution per minute • Physical mechanism • perpendicular wall reaction reorients bacteria motion • either bacteria slides to corner − → gets stuck − → torque or bacteria slides away from corner − → no torque white "head" : self-propulsion direction Bauer & Cornu : First passage FR & cycle affinities Semi-Markovian processes Firenze, 2014/05/30 3 / 30
1.2 Modelization by a finite state semi-Markovian process • Finite number of configurations C m : discretized values of angle α of black spot position : C m ≡ α m = m 2 π/ M • Semi-Markovian process (or generalized renewal sequence ) : � � ( C 0 , τ 0 ) , ( C , τ 0 + τ ) , ( C ′ , τ 0 + τ + τ ′ ) , . . . History : After a waiting time τ distributed with probability P C ( τ ) , system jumps from C to C ′ with probability ( C ′ | P |C ) ( P stochastic matrix with quantum mechanics convention for sense of evolution) Bauer & Cornu : First passage FR & cycle affinities Semi-Markovian processes Firenze, 2014/05/30 4 / 30
1.2 Modelization by a finite state semi-Markovian process • Finite number of configurations C m : discretized values of angle α of black spot position : C m ≡ α m = m 2 π/ M • Semi-Markovian process (or generalized renewal sequence ) : � � ( C 0 , τ 0 ) , ( C , τ 0 + τ ) , ( C ′ , τ 0 + τ + τ ′ ) , . . . History : After a waiting time τ distributed with probability P C ( τ ) , system jumps from C to C ′ with probability ( C ′ | P |C ) ( P stochastic matrix with quantum mechanics convention for sense of evolution) • Graph representation : � configuration C vertex • : weight for waiting time at C : - P 0 C ( τ ) if C initial configuration of history - P C ( τ ) otherwise bond —– : probability ( C ′ | P |C ) to jump from C to C ′ when a jump is known to occur and probability ( C| P |C ′ ) of reverse jump Bauer & Cornu : First passage FR & cycle affinities Semi-Markovian processes Firenze, 2014/05/30 4 / 30
1.3 Questions 1) Probability that the cycle be performed at least once in positive (negative) sense in a infinite time interval ? 2) Fluctuation relation for first passage time at winding number +1 or -1 ? winding number = number of revolutions in the positive sense minus number of revolutions in the opposite sense Answers use affinity concept Bauer & Cornu : First passage FR & cycle affinities Semi-Markovian processes Firenze, 2014/05/30 5 / 30
AFFINITY and ENTROPY PRODUCTION RATE Known results for Markovian processes Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 6 / 30
2.1 Specific case : Markovian processes • Markov property : specific form for probability of waiting time τ in configuration C : exponential P C ( τ ) = r ( C ) e − r ( C ) τ r ( C ) escape rate from C = inverse mean waiting time at C • From a Markov chain to a Markov process : ( C ′ | P |C ) probability to jump from C to C ′ knowing that system jumps out of C → ( C ′ | W |C ) dt probability to jump from C to C ′ during dt − • Master equation for evolution of probability P ( C ; t ) of configuration C at t � dP ( C ; t ) [( C| W |C ′ ) P ( C ′ ; t ) − ( C ′ | W |C ) P ( C ; t )] = dt C ′ � = C • Microreversibility hypothesis : ( C ′ | W |C ) � = 0 ( C| W |C ′ ) � = 0 ⇔ Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 7 / 30
2.2 Shannon-Gibbs entropy evolution and irreversibility • Dimensionless Shannon-Gibbs entropy ( k B = 1) � SG [ P ( t )] ≡ − S P ( C ; t ) ln P ( C ; t ) C � dS SG ( C ′ | W |C ) P ( C ; t ) ln P ( C ; t ) = P ( C ′ ; t ) dt C , C ′ Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 8 / 30
2.2 Shannon-Gibbs entropy evolution and irreversibility • Dimensionless Shannon-Gibbs entropy ( k B = 1) � SG [ P ( t )] ≡ − S P ( C ; t ) ln P ( C ; t ) C � dS SG ( C ′ | W |C ) P ( C ; t ) ln P ( C ; t ) = P ( C ′ ; t ) dt C , C ′ • Analogy with phenomenological thermodynamics of irreversible processes [Schnakenberg 1976] dS SG = d exch S SG + d irr S SG dt dt dt � ( C ′ | W |C ) P ( C ; t ) ln ( C ′ | W |C ) d exch S SG ≡ − with no definite sign ( C| W |C ′ ) dt C , C ′ � [( C ′ | W |C ) P ( C ; t ) − ( C| W |C ′ ) P ( C ′ ; t )] ln ( C ′ | W |C ) P ( C ; t ) d irr S SG ≡ 1 ( C| W |C ′ ) P ( C ′ ; t ) ≥ 0 dt 2 C , C ′ d irr S SG : irreversible entropy production rate dt Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 8 / 30
2.3 Comparison with kinetic theory : affinity of a chemical reaction (a) • In a vessel with walls at inverse temperature β and exerting pressure P , one introduces species A and B prepared separately at ( β, P ) reversible reaction : A ⇋ B • Phenomenological thermodynamics of irreversible processes d irr S ph dn A ⇋ B B = β ( µ A − µ B ) × � �� � dt dt � �� � � �� � entropy production rate affinity A A ⇋ B reaction extent rate J A ⇋ B µ i chemical potential ( i = A , B , n i : molecule concentration for species i ) Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 9 / 30
2.3 Comparison with kinetic theory : affinity of a chemical reaction (a) • In a vessel with walls at inverse temperature β and exerting pressure P , one introduces species A and B prepared separately at ( β, P ) reversible reaction : A ⇋ B • Phenomenological thermodynamics of irreversible processes d irr S ph dn A ⇋ B B = β ( µ A − µ B ) × � �� � dt dt � �� � � �� � entropy production rate affinity A A ⇋ B reaction extent rate J A ⇋ B µ i chemical potential ( i = A , B , n i : molecule concentration for species i ) • Kinetic theory : dn A ⇋ B B = k B ← A n A − k A ← B n B with k j ← i : kinetic constants dt • Thermodynamics of ideal solutions : n i ∝ e βµ i and µ eq A = µ eq B → β ( µ A − µ B ) = ln k B ← A n A k A ← B n B Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 9 / 30
2.3 Comparison with kinetic theory : affinity of a chemical reaction (b) • Correspondance: concentration n i ( t ) − → P ( C ; t ) configuration probability → ( C ′ | W |C ) transition rate kinetic constant k j ← i − � − → Rewriting d irr S SG = 1 J C ⇋ C ′ A C ⇋ C ′ dt 2 C , C ′ J C ⇋ C ′ ≡ ( C ′ | W |C ) P ( C ; t ) − ( C| W |C ′ ) P ( C ′ ; t ) bond current A C ⇋ C ′ ≡ ln ( C ′ | W |C ) P ( C ; t ) bond affinity ( C| W |C ′ ) P ( C ′ ; t ) Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 10 / 30
2.4 Affinity for a master equation corresponding to a graph made of a single cycle • Representation of a master equation by a graph Graph G : vertex • : configuration C bond —– : transtion rates ( C ′ | W |C ) and ( C| W |C ′ ) • Case where graph G is a cycle C of M vertices . Fixed orientation along C with C M + 1 ≡ C 1 M � ( C m + 1 | W |C m ) P ( C m ; t ) cycle affinity A C ≡ A C m ⇋ C m + 1 with A C m ⇋ C m + 1 ≡ ln ( C m | W |C m + 1 ) P ( C m + 1 ; t ) m = 1 M � ( C m + 1 | W |C m ) A C = ln independent from P ( C , t ) ( C m | W |C m + 1 ) m = 1 Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 11 / 30
2.4 Affinity for a master equation corresponding to a graph made of a single cycle • Representation of a master equation by a graph Graph G : vertex • : configuration C bond —– : transtion rates ( C ′ | W |C ) and ( C| W |C ′ ) • Case where graph G is a cycle C of M vertices . Fixed orientation along C with C M + 1 ≡ C 1 M � ( C m + 1 | W |C m ) P ( C m ; t ) cycle affinity A C ≡ A C m ⇋ C m + 1 with A C m ⇋ C m + 1 ≡ ln ( C m | W |C m + 1 ) P ( C m + 1 ; t ) m = 1 M � ( C m + 1 | W |C m ) A C = ln independent from P ( C , t ) ( C m | W |C m + 1 ) m = 1 • Property of stationary state P st ( C ) Cycle current : J C [ P st ] ≡ J C 1 ⇋ C 2 [ P st ] = J C 2 ⇋ C 3 [ P st ] = · · · � � d irr S SG � Entropy production rate: = J C [ P st ] A C � dt P st Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 11 / 30
2.5 Affinity class in graph theory • Exchange processes in configuration jumps ↔ antisymmetric matrices - S for the exchange entropy variation - A for the affinity variation ( C ′ | S |C ) ≡ ln ( C ′ | W |C ) ( C ′ | A [ P ] |C ) ≡ ln ( C ′ | W |C ) P ( C ; t ) and ( C| W |C ′ ) P ( C ′ ; t ) ≡ A C ⇋ C ′ ( C| W |C ′ ) Bauer & Cornu : First passage FR & cycle affinities Affinity Firenze, 2014/05/30 12 / 30
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