Stochastic Thermodynamics with Martingales Izaak Neri, Workshop on - - PowerPoint PPT Presentation

13 Mar 2018

Izaak Neri, Workshop on Martingales 
 in Finance and Physics, 24th of May 2019

Stochastic Thermodynamics with Martingales

Contributions

Edgar Roldán (Trieste) Frank Jülicher (Dresden) Simone Pigolotti (Okinawa) Raphaël Chétrite (Nice) Shamik Gupta (Calcutta)

Statistical physics

Meik Dörpinghaus (Dresden)

Information theory

Heinrich Meyr (Aachen)

Structure of the talk

2.Exponential martingale structure of entropy production 3.Thermodynamic laws 
 at stopping times

• 4. Universal properties 

• f entropy production
• 5. Example: overdamped Langevin processes
• 1. Introduction to stochastic thermodynamics

Introduction to stochastic thermodynamics

diffusive systems colloidal particles


Thermodynamics of mesoscopic systems or stochastic thermodynamics

Thermodynamics:

diffusive systems colloidal particles


Thermodynamics of mesoscopic systems or stochastic thermodynamics

Thermodynamics: Stochastic 
 thermodynamics:

+ +

=

+

=

U Seifert, Rep. Prog. Phys. (2012 )

Local detailed balance and stochastic entropy production

U Seifert, Rep. Prog. Phys. (2012 )

Local detailed balance and stochastic entropy production

where

Thermodynamic laws for mesoscopic processes

Integral fluctuation relation:

Events of negative entropy production must exist Implications

Thermodynamic laws for mesoscopic processes

Integral fluctuation relation:

U Seifert, Rep. Prog. Phys. (2012 )

Exponential martingale structure 


• f entropy production

Martingales

M(t) is a martingale with respect to X(t) if: M(t) is a real-valued function on X(0…t) , for all s<t

IN, Edgar Roldan, Frank Julicher, Phys. Rev. X (2017)

For stationary processes the exponential of the negative entropy production is a martingale

IN, Edgar Roldan, Frank Julicher, Phys. Rev. X (2017)

For stationary processes the exponential of the negative entropy production is a martingale

IN, Edgar Roldan, Frank Julicher, Phys. Rev. X (2017)

= X

X(s+...t)

p (X(0 . . . t)) p (X(0 . . . s)) ˜ p (X(0 . . . t)) p (X(0 . . . t))

For stationary processes the exponential of the negative entropy production is a martingale

IN, Edgar Roldan, Frank Julicher, Phys. Rev. X (2017)

= X

X(s+...t)

p (X(0 . . . t)) p (X(0 . . . s)) ˜ p (X(0 . . . t)) p (X(0 . . . t))

For stationary processes the exponential of the negative entropy production is a martingale

IN, Edgar Roldan, Frank Julicher, Phys. Rev. X (2017)

= X

X(s+...t)

p (X(0 . . . t)) p (X(0 . . . s)) ˜ p (X(0 . . . t)) p (X(0 . . . t))

For stationary processes the exponential of the negative entropy production is a martingale

IN, Edgar Roldan, Frank Julicher, Phys. Rev. X (2017)

= X

X(s+...t)

p (X(0 . . . t)) p (X(0 . . . s)) ˜ p (X(0 . . . t)) p (X(0 . . . t))

For stationary processes the exponential of the negative entropy production is a martingale

Thermodynamic laws at stopping times

Can a gambler make fortune in a fair game by quitting at an intelligently chosen moment?

2 4 6 8 10

• 1.5
• 1
• 0.5

0.5 1 1.5 1 2 3 4

• 1.5
• 1
• 0.5

0.5 1 1.5

Gambler makes profit Gambler on average makes no profit

Can a gambler make fortune in a fair game by quitting at an intelligently chosen moment?

No, if the gambler cannot foresee the future, cannot cheat, 
 and has access to a finite budget

Can a gambler make fortune in a fair game by quitting at an intelligently chosen moment?

No, if the gambler cannot foresee the future, cannot cheat, 
 and has access to a finite budget T is a stopping time

Can a gambler make fortune in a fair game by quitting at an intelligently chosen moment?

No, if the gambler cannot foresee the future, cannot cheat, 
 and has access to a finite budget T is a stopping time M(t) is uniformly integrable

Can a gambler make fortune in a fair game by quitting at an intelligently chosen moment?

No, if the gambler cannot foresee the future, cannot cheat, 
 and has access to a finite budget T is a stopping time M(t) is uniformly integrable if M(t) is uniformly integrable martingale and

R S Lipster and A N Shiryaev, Statistics of random processes: I General theory, 1977

and T is a stopping time Doob’s optional stopping theorem

IN, E Roldan, S Pigolotti, F Julicher, arXiv (2019)

Integral fluctuation relations for entropy production at stopping times

Finite time windows

IN, E Roldan, S Pigolotti, F Julicher, arXiv (2019)

Integral fluctuation relations for entropy production at stopping times

Finite time windows Infinite time windows

IN, E Roldan, S Pigolotti, F Julicher, arXiv (2019)

if and

Integral fluctuation relations for entropy production at stopping times

Second law of thermodynamics at stopping times

Jensen’s Inequality

Second law of thermodynamics at stopping times

Jensen’s Inequality

Second law of thermodynamics at stopping times

Jensen’s Inequality For isothermal processes:

Universal properties of entropy production

Universal properties of entropy production (for continuous stochastic processes)

Time Entropy production

b)

• =

Splitting probabilities of the entropy production 


Time Entropy production

b)

• =

Splitting probabilities of the entropy production 


Time Entropy production

b)

• =

Splitting probabilities of the entropy production 


Time Entropy production

b)

• =

Splitting probabilities of the entropy production 


Entropy production Time

−1

The statistics of minima of the entropy production

• f continuous stationary processes are universal

IN, Edgar Roldán, Frank Jülicher, Phys. Rev. X 7, 011019

Entropy production Time

−1

The statistics of minima of the entropy production

• f continuous stationary processes are universal

IN, Edgar Roldán, Frank Jülicher, Phys. Rev. X 7, 011019

Entropy production Time

−1

The statistics of minima of the entropy production

• f continuous stationary processes are universal

IN, Edgar Roldán, Frank Jülicher, Phys. Rev. X 7, 011019

Bounds on negative fluctuations of entropy production: “standard” thermodynamics vs martingale theory

U Seifert, Rep. Prog. Phys. (2012 ) IN, E Roldan, S Pigolotti, F Julicher, arXiv (2019)

Symmetry relation in the conditional distributions 


• f first-passage times for entropy production

Time Entropy production

b)

• =

IN, Edgar Roldán, Frank Jülicher, Phys. Rev. X 7, 011019 Meik Dorpinghaus, IN, Edgar Roldan, Frank Julicher, Heinrich Meyer, arXiv

Symmetry relation in the conditional distributions 


• f first-passage times for entropy production

IN, Edgar Roldán, Frank Jülicher, Phys. Rev. X 7, 011019 Meik Dorpinghaus, IN, Edgar Roldan, Frank Julicher, Heinrich Meyer, arXiv

Duality in first-passage times

Duality in first-passage times

Continuous
 processes

Processes 
 with jumps

???

Continuous
 processes

Example: overdamped Langevin processes

System set-up ,

First law of thermodynamics for overdamped Langevin process

System set-up ,

First law of thermodynamics for overdamped Langevin process

Ito product

System set-up ,

First law of thermodynamics for overdamped Langevin process

Ito product

First law of thermodynamics

,

K Sekimoto, Prog. Theory. Phys. Suppl. 130, 17 (1998)

System set-up ,

First law of thermodynamics for overdamped Langevin process

Ito product

First law of thermodynamics

,

Stratanovich 
 product

K Sekimoto, Prog. Theory. Phys. Suppl. 130, 17 (1998)

Second law of thermodynamics for

• verdamped Langevin process

Definition of entropy production: where

Udo Seifert, Physical review letters (2005)

Second law of thermodynamics for

• verdamped Langevin process

Definition of entropy production: where

Udo Seifert, Physical review letters (2005)

Second law of thermodynamics for

• verdamped Langevin process

Rules of stochastic calculus imply: Definition of entropy production: where

Udo Seifert, Physical review letters (2005)

, ,

Second law of thermodynamics for

• verdamped Langevin process

Rules of stochastic calculus imply: Definition of entropy production: where

Udo Seifert, Physical review letters (2005)

, ,

Second law of thermodynamics for

• verdamped Langevin process

Rules of stochastic calculus imply: , , Definition of entropy production: where

Udo Seifert, Physical review letters (2005)

Exponential martingale structure of entropy production

• S. Pigolotti, IN, E. Roldán, and F. Jülicher, Phys. Rev. Lett. 119, 140604

,

Exponential martingale structure of entropy production

• S. Pigolotti, IN, E. Roldán, and F. Jülicher, Phys. Rev. Lett. 119, 140604

,

Exponential martingale structure of entropy production

• S. Pigolotti, IN, E. Roldán, and F. Jülicher, Phys. Rev. Lett. 119, 140604

,

Exponential martingale structure of entropy production

• S. Pigolotti, IN, E. Roldán, and F. Jülicher, Phys. Rev. Lett. 119, 140604

,

Exponential martingale structure of entropy production

• S. Pigolotti, IN, E. Roldán, and F. Jülicher, Phys. Rev. Lett. 119, 140604

No drift term —-> martingale

,

Random-time transformation

• S. Pigolotti, IN, E. Roldán, and F. Jülicher, Phys. Rev. Lett. 119, 140604

Entropic time:

Entropy production Time

Nondecreasing Martingale

,

Random-time transformation renders certain properties of entropy production universal

• S. Pigolotti, IN, E. Roldán, and F. Jülicher, Phys. Rev. Lett. 119, 140604

Fluctuation properties independent of 
 the time-scale are universal

0.02 0.04 0.06 20 40 60

t=1

Probability density

probability distribution

Inequality for the Fano-factor of entropy production

0.02 0.04 0.06 20 40 60

t=1

Probability density

probability distribution

Inequality for the Fano-factor of entropy production

0.02 0.04 0.06 20 40 60

t=1

Probability density

probability distribution

Inequality for the Fano-factor of entropy production

Fano factor inequality for entropy production

, ,

Fano factor inequality for entropy production

, ,

= 2 Ito isometry

Fano factor inequality for entropy production

, ,

= 2 Ito isometry = 0 Doob h-transform

Fano factor inequality for entropy production

Discussion

• Entropy production has an exponential martingale structure

Discussion

• Entropy production has an exponential martingale structure
• Thermodynamic laws at stopping times, 


which may be first-passage times

Discussion

• Entropy production has an exponential martingale structure
• Thermodynamic laws at stopping times, 


which may be first-passage times

• Universal fluctuations properties of entropy production: 


infimum statistics, splitting probabilities, etc.

Discussion

• Entropy production has an exponential martingale structure
• Thermodynamic laws at stopping times, 


which may be first-passage times

• Universal fluctuations properties of entropy production: 


infimum statistics, splitting probabilities, etc. Thank you for your attention!