Information Thermodynamics on Causal Networks

Transcription

1 1/39 Information Thermodynamics on Causal Networks FSPIP 2013, July Sosuke Ito Dept. of Phys., the Univ. of Tokyo (In collaboration with T. Sagawa) ariv:

2 The second law of thermodynamics in small systems 2/39 ΔS total = ΔS bath + ΔS system = 0 Q k T B Δ log p system :Entropy production (stochastic) The heat bath with temperature T Small system ΔS bath = Q k T B Heat Q Control describes the ensemble average. ΔSsystem = Δln p system Shannon entropy change

3 3/39 Nonequilibrium equalities The fluctuation theorem p B p( = ω) ( = ω) = exp( ω) The detailed fluctuation theorem p( xˆ( t) x(0)) ln * * p ( xˆ ( τ )) B = β Q i i i t) x ( τ p B :The probability of the backward process { x( t) τ } xˆ ( t) = 0 t * :Path :Time- reversed The integrated fluctuation theorem, IFT (the Jarzynski equality) exp( ) = 1 (D. J. Evans, E. G. D. Cohen and G. P. Morriss. PRL. 71, 2401, 1993.) (C. Jarzynski. PRL. 78, 2690, 1997.) The second law 0 can be derived from these relations.

4 In the case where the system is coupled to multiple small systems 4/39?, Y? Q k T B Δ ln p The heat bath with temperature T Heat Q Small system Heat Y Q interactions Small system Y

5 In the case where Y seems to be Maxwell s demon 5/39 (information entropy)? Q k T B Δ ln p Leó Szilárd (1929) Léon Brillouin (1951) CH. BenneN (1982) The heat bath with temperature T Heat Q Measurement Small system Heat Q Y Feedback control Small system Y

6 6/39 In the context of the fluctuation theorems The generalized IFT: [ ] + ΔI 1 exp = Sagawa- Ueda: ΔI (T. Sagawa and M. Ueda. PRL 104, , 2010) (T. Sagawa and M. Ueda. PRL 109, , 2012) Δ I :The mutual information that is exchanged between the system and the demon Y ΔI ΔH + ΔH Y ΔH Y H = p()ln p() H Y = p(y )ln p(y ) H Y = p(,y )ln p(,y ),Y Y

7 7/39 Relevant studies In the case of the repeated feedback control Y. Fujitani and H. Suzuki, JPSJ 79, , J. M. Horowitz and S. Vaikuntanathan, PRE 82, , T. Sagawa and M. Ueda, PRE 85, , S. Lahiri, S. Rana and A. M. Jayannavar, J. Phys. A: Math. Theor. 45, , The (semi) quantum generalized IFT Y. Morikuni and H. Tasaki, J. Stat. Phys 143, 1, The generalized IFT in NESS D. Abreu and U. Seifert, PRL 108, , Experimental demonstration of Maxwell demon S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki and M. Sano Nature Physics 6, /

8 The general theory has been elusive for more complex cases 8/39 Θ Can we generally obtain such an inequality? What is the physical meaning of Θ? The heat bath with temperature T Heat Q Small system Q k T B Δ ln p

9 The general theory has been elusive for more complex cases 9/39 Θ Can we generally obtain this inequality? What is the physical meaning of Θ? Q k T B Δ ln p Small system Brownian particles with interactions

10 10/39 How to create a general theory We describe nonequlibrium dynamics of multiple systems. We obtain Θ for such systems. [ + Θ], exp =1 Θ (In the case of Maxwellʼ s demon, this result reproduces Θ=ΔI.)

11 11/39 How to create a general theory We describe nonequlibrium dynamics of multiple systems. Graphical representation by causal networks We obtain Θ for such systems. [ + Θ], exp =1 Θ (In the case of Maxwellʼ s demon, this result reproduces Θ=ΔI.)

12 12/39 How to create a general theory We describe nonequlibrium dynamics of multiple systems. We obtain Θ for such systems. [ + Θ], exp =1 (In the case of Maxwellʼ s demon, this result reproduces Θ=ΔI.) Θ Graphical representation by causal networks Θ is the informatic term characterized by the topology of the graph.

13 13/39 Causal networks A probabilistic graphical model that represents their conditional dependencies We can describe the complex interactions Causality (represented by directed acyclic graphs) We can describe the time ordering by the topology of the graph (topological ordering). Quite a broad class of stochastic processes Markov chain, Non Markov process, multidimensional Langevin dynamics, Poisson process etc. F. V. Jensen, Bayesian Networks and Decision Graphs. (Springer, Berlin, 2001) Yesterday H. Touchette discussed the relationship between Shannon entropy and mutual information using this networks (DAG). [H ToucheNe, S Lloyd - Physica A, 2004]

14 14/39 Causal relationships a 1 a2,, a 3 : Random variables Causal relationships are described by the directed edges. a 2 a 3 a 1 In this case, we call that are parents of a 1 and a 2 The set of parents of a 3 : a 3. pa( a ) 3 = { a 1, a 2 }

15 15/39 Examples of Bayesian networks y 1 z 1 z 0 x 0 x 0 y 0 pa( ) = {, x 0 } pa( ) = {x 0 } pa(x 0 ) = (Empty set) pa( ) = {, y 0 } pa( ) = {x 0, y 0 } pa(y 1 ) = {, x 0, y 0, z 0 }

16 16/39 The counter example y 1 z 1 z 0 x 0 x 0 y 0 The graph should be acyclic because of the causality.

17 17/39 Topological ordering a 1 a2,, a 3 : Random variables a 3 A linear ordering of variables such that node a j cannot be a parent of a jʼ with j>jʼ. a 2 a 1 a 1, a 2, a 3 is an example of topological ordering in this graph. a 2, a 1, a 3 is also topological ordering.

18 18/39 Topological ordering y 1 z 1 z 0 x 0 x 0 y 0 x 0, y 0,, z 0, y 1, z 1, x 0,, y 0, x 0,, z 0,, y 1, z 1

19 19/39 Characterizing causal relationships a 1 a2,, a a 2 3 : Random variables a 3 a 1 By the conditional probability ( ( )) = p a j a j 1,, a 1 p a j pa a j The chain rule ( ) = p a j pa a j p a 1,, a n n j=1 pa(a 1 ) = pa(a 2 ) = φ { } ( ) Topological ordering ( ( )) (empty set) pa(a 3 ) = a 1, a 2 p ( a1, a2, a3) = p( a3 a1, a2) p( a2) p( a1)

20 20/39 Characterizing causal relationships y 1 z 1 z 0 x 0 x 0 y 0 p(x 0,, ) = p(, x 0 )p( x 0 )p(x 0 ) p(x 0,,, y 0, y 1, z 0, z 1 ) = p(z 1 y 1, z 0 )p(y 1 z 0,, y 0 )p(, y 0 ) p(z 0 )p( x 0 )p(y 0 )p(x 0 )

21 An example of Bayesian network that describe a stochastic process 21/39 Markov chain x 3 The state of the system x i x( t = iδt) The path probability p( x1 ) p( x2 x1 ) p( x3 x2)

22 An example of Bayesian network that describe a stochastic process 22/39 Multidimensional Langevin dynamics x 3 y 3 y 2 y 1 z 3 z 2 z 1 x i+1 = x i + f x x i x( t = idt) ( x i, y i )dt +ξ x dt y i y( t = idt) z i z( t = idt) dx = f x dt dy = f y dt = x, y, z dz = f z ξ (t) = 0, dt ξ ( t)ξ ' (t ') = D δ ' δ(t t ') ( x, y) ( y, z) + ξ + ξ ( z, x) + ξ z x y

23 An example of Bayesian network that describe a stochastic process 23/39 Szilard engine The path probability p( x1 ) p( m1 x1 ) p( x2 m1, x1 ) m 1 m 1 Szilard engine

24 An example of Bayesian network that describe a stochastic process 24/39 Szilard engine The path probability p( x1 ) p( m1 x1 ) p( x2 m1, x1 ) m 1 m 1 Measurement Szilard engine

25 An example of Bayesian network that describe a stochastic process 25/39 Szilard engine The path probability p( x1 ) p( m1 x1 ) p( x2 m1, x1 ) Time evolution under feedback control Feedback m 1 Measurement m 1 Szilard engine

26 How to create 26/39 a general theory We would like to obtain the bound Θ. [ + Θ], exp =1 Θ To calculate Θ on Bayesian networks 1:The definition of ensemble average The joint probability is given. ( ) = p a j pa a j p a 1,, a n n j=1 ( ( )) f p a 1,, a n a 1,,a n 2:The definition of entropy production ( ) f ( a 1,, a n )

27 Entropy production in the system The entropy production in x s on Bayesian networks = β i Q i i :the final state Δ ln p s 1 ln p(x x, B ) j+1 j j+1 p B (x j x j+1, B j+1 ) + ln p(x t ) p(x s ) j=t Detailed fluctuation theorem x t :the initial state Time evolution x j+1 27/39 p B :The probability of the backward process Contribution (i.e., memory) Our definition is equivalent to the Sekimotoʼ s definition of the heat for the Langevin system up to o(dt). x j B j+1 pa(x j+1 ) \ { x j }

28 28/39 Main result The novel generalization of the second law and the IFT. [ + Θ], exp =1 Θ Θ = I fin I ini l I l tr I ini : Initial correlation between the target system and other systems I fin : Final correlation between the system and others I trl : Information transfer from the system to others during the dynamics

29 29/39 I ini : Initial correlation I ini : Initial correlation between the system and other systems I ini ln p(, pa( )) p( )p(pa( )) x N pa( )

30 30/39 I fin : Final correlation I fin : Final correlation between the system and others I fin ln p(x N,C) p(x N )p(c) x N c N ' Topological ordering { } \,, x N C a 1,, a j with a j = x N C is the set of random variables that effect on the final state from outside of. { } x N 1 c N ' 2 c N ' 3 c N ' 1 c N ' 4 C

31 31/39 I l tr : Information transfer I tr : Information transfer from into c l during the dynamics (l =1,, N) l I tr ln p(pa p(c l c l 1,, c 1 )p(pa ( c l ),c l c l 1,,c 1 ) c l ( ) c l 1,,c 1 ) This term is equivalent to the entropy transfer in Ref. [T. Schreiber, PRL 85, 461, 2000.] { } C = c l 1 l N ' Topological ordering pa (c l ) pa(c l ) {,, x N } x N x N 1 c N ' 2 c N ' 3 c N ' c N ' 1 c N ' 4 C

32 32/39 Example 1: Markov chain On this Bayesian network, C =φ pa( ) = φ x N I fin = 0 I = 0 ini l I tr :none 0 exp( ) =1

33 33/39 Example 2: Feedback control On this Bayesian network, x N C = ( ) I = I x : m fin N 1 I ini = 0 { } m 1 pa( ) = φ m 1 1 tr I = I I ( m : x ) 1 1 ( x : m ) I( x m ) N 1 1 : and the generalized IFT. 1

34 34/39 Example 3: Two-dimensional Langevin system This is a NEW result. x N y N y 2 y 1 On this Bayesian network, C = y, y 1 N pa(x ) = φ 1 I I ( x { y y }),, fin = I : N 1 N 1 I ini = 0 ( y : x y y ) l = I,, tr l+ 1 l l { }, 1 I fin N 1 l= 1 I l tr and the generalized IFT. 1

35 35/39 Example 4: Repeated feedback control x N m 1 m 2 m N 1 On this Bayesian networks, C = m, m 1 N pa(x ) = φ 1 I I ( x { m m }),, fin I ini = 0 { } = I : N 1 N 1 ( m : x m m ) l = I,, tr l l l 1, 1 I fin N 1 l= 1 I l tr and the generalized IFT. 1

36 36/39 Example 4: Repeated feedback control x N m N 1 J. M. Horowitz and S. Vaikuntanathan, PRE 82, , N 1 βw d l I tr l=1 m 2 If the system is initially and finally in equilibrium, we have I fin = βw d m 1 Our result on this Bayesian network is equivalent to the result obtained by Horowitz Vaikuntanathan.

37 Example 5: Time-delayed feedback control This is a NEW result. On this Bayesian networks, { } C = m 1, m 2 37/39 m 2 m 1 pa(x ) = m 1 1 I fin = I : m 1, m 2 I ini = I : m 1 ( { }) ( ) ( ) I tr 2 = I m 2 : m 1 1 I tr = 0 I ( :{ m 1, m }) 2 I ( : m ) 1 I ( m 2 : m ) 1 and the generalized IFT.

38 38/39 Example 5: Time-delayed feedback control m 2 The three- body Shannon entropy change that includes the states of different times m 1 and m 2. m 1 This is a crucial difference between the conventional thermodynamics and our result. The result can be rewritten as β i Q i i ln p, m 1, m 2 ( ) + ln p(, m 1, m ) 2 0

39 39/39 Conclusion We derive the novel generalizations of the IFT and the second law in the presence of a complex information flow between multiple nonequilibrium system. exp( + Θ) =1 Θ Θ = The new informatic term Θ is characterized by the topology on the causal structure of dynamics. Our theory is applicable to quite a broad class of nonequilibrium dynamics such as multidimensional Langevin dynamics and time- delayed feedback control, which may occur in biochemical reactions (i.e., adaptation). I fin I ini l I l tr