Stochastic thermodynamics of information

Introduction Textbook, Review Sekimoto, K. (2010). Stochastic energetics (Vol. 799). Springer. Seifert, U. (2012). Reports on Progress in Physics, 75(12), 126001.

Introduction Review Parrondo, J. M., Horowitz, J. M., & Sagawa, T. (2015). Thermodynamics of information. Nature physics, 11(2), 131. My Ph, D. thesis Ito, S. (2016). Information thermodynamics on causal networks and its application to biochemical signal transduction. Springer. Toyabe, S., Sagawa, T., Ueda, M., Muneyuki, E., & Sano, M. (2010). Nature physics, 6(12), 988.

Introduction Ito, S., & Sagawa, T. (2013). Physical review letters, 111(18), 180603. Shiraishi, N., Ito, S., Kawaguchi, K., & Sagawa, T. (2015). New Journal of Physics, 17(4), 045012. Ito, S. (2016). Scientific reports, 6. Ito, S., & Sagawa, T. (2015). Nature communications, 6. Yamamoto, S., Ito, S., Shiraishi, N., & Sagawa, T. (2016). Physical Review E, 94(5), 052121. Ito, S. (2017). arxiv preprint arxiv:1712.04311.

Introduction

dq = du dw dw = @ U d dq du dw λ. dq ds T ds dq T

dq ds T ds bath = dq T ds + ds bath 0

m X ẍ(t) = X ẋ(t) @ x U X (x(t), X (t)) + p 2 X T X X (t) ξx x(t) t mx γx UX λx TX mx γx γx=1 ẋ(t) = @ x U X (x(t), X (t)) + p 2T X X (t)

dux. du X (x(t), X (t)) = ẋ(t) @ x U X (x(t), X (t))dt + X (t) @ X U X (x(t), X (t))dt dt dwx dqx dw X (x(t), X (t)) = X (t) @ X U X (x(t), X (t))dt dq X (x(t), X (t)) = ẋ(t) @ x U X (x(t), X (t))dt dq X = du X dw X

dq X (x(t), X (t)) = ẋ(t) @ x U X (x(t), X (t))dt ẋ(t) = @ x U X (x(t), X (t)) + p 2T X X (t) dq X =ẋ(t) ( p 2T X X (t) ẋ(t))dt jx=dqx/dt j X (t) =ẋ(t) ( p 2T X X (t) ẋ(t))

X. x. px(x) X Z H(X) = dxp X (x)lnp X (x) =h ln p X (x)i px1,,xn (x1,, xn) H(X 1,...,X n )= Z dx 1 dx n p X1,...,X n (x 1,...,x n )lnp X1,...,X n (x 1,...,x n ) = h ln p X1,...,X n (x 1,...,x n )i

Y X H(X Y )=H(X, Y ) H(Y ) X, Y I(X; Y )=H(X) H(X Y ) Z X Y I(X; Y Z) =H(X Z) H(X Y,Z)

I(X; Y )=I(Y; X) I(X; Y ) 0 I(X; Y )=0 p X,Y (x, y) =p X (x)p Y (y) x, y X Y 0 I(X; Y ) H(X Y ) H(Y X) H(X) H(Y ) I(X; Y )=H(X)+H(Y) H(X, Y ) = H(X) H(X Y ) = H(Y ) H(Y X)

px(x), qx(x) D(p X q X )= Z dxp X (x)ln p X(x) q X (x) I(X; Y )=D(p X,Y p X p Y )

D(p X q X ) 0 D(p X q X )=0 p X (x) =q X (x) x F, x G(x), px(x) Z Z F (G(x))p X (x)dx apple F G(x)p X (x)dx hf (G(x))i applef (hg(x)i) F=ln G(x)= qx(x)/px(x), D(p X q X )=hln[q X (x)/p X (x)]i appleln[hq X (x)/p X (x)i] =ln1=0

ẋ(t) = @ x U X (x(t), X (t)) + p 2T X X (t) X (t)dt = db X t x t = x(t) X t = X (t) x t+dt x t = @ x U X (x t, X t)dt + p 2T X db X t dbxt p(db Xt )= 1 p 2 dt exp apple (dbxt ) 2 2dt @[db X t] @x t+dt = 1 p 2TX xt xt+dt T (x t+dt ; x t )=p Xt+dt X t = 1 p 4 TX dt exp apple (xt+dt x t + @ x U X (x t, X t )dt) 2 4T X dt

xt+dt xt. p B X t X t+dt (x t x t+dt )=T (x t ; x t+dt ) apple p Xt+dt X t (x t+dt x t ) p B X t X t+dt (x t x t+dt ) =exp jx (t) T X dt T (x t+dt ; x t )=p Xt+dt X t = apple 1 p 4 TX dt exp (xt+dt x t + @ x U X (x t, X t )dt) 2 4T X dt j X (t) =ẋ(t) ( p 2T X X (t) ẋ(t))

D(p Xt+dt X t p Xt p B X t X t+dt p Xt+dt ) 0 D(p Xt+dt X t p Xt p B X t X t+dt p Xt+dt )= hj X (t)idt T X + H(X t+dt ) H(X t ) X ds X (t) =H(X t+dt ) H(X t ) ds bath (t) = hj X (t)i T X dt ds X (t)+ds bath (t) 0

λx m. ẋ(t) = @ x U X (x(t), X (m, t)) + p 2T X X (t) m T m (x t ; x t+dt )=p Xt+dt X t,m (x t+dt x t,m)

m p B X t X t+dt,m (x t x t+dt,m)=t m (x t ; x t+dt ) p Xt+dt X t,m (x t+dt x t,m) p B X t X t+dt,m (x t x t+dt,m) =exp apple jx (t) T X dt j X (t) =ẋ(t) ( p 2T X X (t) ẋ(t))

D(p Xt+dt X t,m p Xt,M p B X t X t+dt,m p Xt+dt,M ) 0 D(p Xt+dt X t,m p Xt,M p B X t X t+dt,m p Xt+dt,M )= X hj X (t)i T X dt + H(X t+dt,m) H(X t,m) ds X (t) =H(X t+dt ) H(X t ) ds bath (t) = ds X (t)+ds bath (t) I(X t+dt ; M) I(X t ; M) hj X (t)i T X dt

Stochastic thermodynamics for 2D Langevin equations ẋ(t) = @ x U X (x(t),y(t)) + p 2T X X (t) ẏ(t) = @ y U Y (x(t),y(t)) + p 2T Y Y (t) ξx ξy p Xt+dt X t,y t (x t+dt x t,y t )=T X y t (x t+dt ; x t ) Ty X t (x t+dt ; x t ) Ty X t (x t ; x t+dt ) =exp apple p Yt+dt X t,y t (y t+dt x t,y t )=T Y x t (y t+dt ; y t ) jx (t)dt T X Tx Y apple t (y t+dt ; y t ) Tx Y t (y t ; y t+dt ) =exp jy (t)dt T Y j X (t) =ẋ(t) ( p 2T X X (t) ẋ(t)) j Y (t) =ẏ(t) ( p 2T Y Y (t) ẏ(t))

Stochastic thermodynamics for 2D Langevin equations D(p Xt+dt X t,y t p Yt+dt X t,y t p Xt,Y t p B X t X t+dt,y t+dt p B Y t X t+dt,y t+dt p Xt+dt,Y t+dt ) 0 D(p Xt+dt X t,y t p Yt+dt X t,y t p Xt,Y t p B X t X t+dt,y t+dt p B Y t X t+dt,y t+dt p Xt+dt,Y t+dt ) = hj X (t)idt T X X Y : hj Y (t)idt T Y + H(X t+dt,y t+dt ) H(X t,y t ) ds X,Y + ds bath,x + ds bath,y 0 ds X,Y = H(X t+dt,y t+dt ) H(X t,y t ) ds bath,x = ds bath,y = hj X (t)idt T X hj Y(t)idt T Y

Stochastic thermodynamics for 2D Langevin equations D(p Xt+dt X t,y t p Xt,Y t,y t+dt p B X t X t+dt,y t+dt p Yt,X t+dt,y) 0 D(p Xt+dt X t,y t p Xt,Y t,y t+dt p B X t X t+dt,y t+dt p Yt,X t+dt,y t+dt ) = hj X (t)idt T X + H(X t+dt ) H(X t )+I(X t ; {Y t,y t+dt }) I(X t+dt ; {Y t,y t+dt }) X: ds X (t) =H(X t+dt ) H(X t ) ds bath,x = hj X (t)idt T X ds X + ds bath,x I(X t+dt ; {Y t,y t+dt }) I(X t ; {Y t,y t+dt })=di

Stochastic thermodynamics for 2D Langevin equations X ds X,Y Y ds X,Y + ds bath,x + ds bath,y 0 hj X idt hj Y idt ds bath,x T X ds bath,y T Y ds X X di Y ds X + ds bath,x di hj X idt hj Y idt ds bath,x T X ds bath,y T Y

Summary