Entropy production due to non-stationary heat conduction

1. Entropy production due to non-stationary heat conduction Ian Ford, Zac Laker and Henry Charlesworth Department of Physics and Astronomy and London Centre for Nanotechnology University College London, UK

2. Three kinds of entropy production • That due to relaxation (cooling of coffee) • That due to maintenance of a steady flow (stirring of coffee; coffee on a hot plate) • That which is left over.... • In this talk I illustrate this separation using a particle in a space- and time-dependent heat bath

3. Stochastic thermodynamics • (Arguably) the best available representation of irreversibility and entropy production

4. Microscopic stochastic differential equations of motion (SDEs) for position and velocity. SDE for entropy change: with positive mean production rate. entropy position time

5. What is entropy change? • We use microscopic equations of motion that break time reversal symmetry. • friction and noise • But what evidence is there of this breakage at the level of a thermodynamic process? • Entropy change is this evidence. • A measure of the preference in probability for a ‘forward’ process rather than its reverse • A measure of the irreversibility of a dynamical evolution of a system

6. Entropy change associated with a trajectory • the relative likelihood of observing reversed behaviour position position time time under forward protocol of driving under reversed protocol

7. Entropy change associated with a trajectory: Sekimoto, Seifert, etc such that In thermal equilibrium, for all trajectories

8. Furthermore! • trajectory entropy production may be split into three separate contributions • Esposito and van den Broek 2010, Spinney and Ford 2012

9. How to illustrate this? • Non-stationary heat conduction

10. Trapped Brownian particle in a non-isothermal medium trap potential: force F(x) = -x temperature position x

11. An analogy: an audience in the hot seats!

12. An analogy: an audience in the hot seats! steady mean heat conduction

13. An analogy: an audience in the hot seats! steady mean heat conduction

14. Stationary distribution of a particle in a harmonic potential well () with a harmonic temperature profile (T) q-gaussian

15. Steady heat current gives rise to entropy production. Now induce production.

16. Steady heat current gives rise to entropy production. Now induce production.

17. Particle explores space- and time-dependent background temperature:

18. Particle probability distribution warm wings

19. Particle probability distribution hot wings

20. Now the maths.....

21. N.B. This probability distribution is a variational solution to Kramers equation • distribution valid in a nearly-overdamped regime • maximisation of the Onsager dissipation functional • which is related to the entropy production rate.

22. and some more maths.... Spinney and Ford, Phys Rev E 85, 051113 (2012) D

23. the remnant.... • only appears when there is a velocity variable • and when the stationary state is asymmetric in velocity • and when there is relaxation

24. Simulations: distribution over position

25. Distribution over velocity at x=0and various t

26. Approx mean total entropy production rate spatial temperature gradient rate of change of temperature Mean ‘remnant’ entropy production is zero at this level of approximation

27. Comparison between average of total entropy production and the analytical approximation

28. Mean relaxational entropy production

29. Mean steady current-related entropy production

30. Distributions of entropy production

31. Some of the satisfy fluctuation relations!

32. Where are we now? • The second law has several faces • new perspective: entropy production at the microscale • Statistical expectations but not rigid rules • Small systems exhibit large fluctuations in entropy production associated with trajectories • Entropy production separates into relaxational and steady current-related components, plus a ‘remnant’ • only the first two are never negative on average • remnant appears in certain underdamped systems only

33. I S Conclusions • Stochastic thermodynamics eliminates much of the mystery about entropy • If an underlying breakage in time reversal symmetry is apparent at the level of a thermodynamic process, its measure is entropy production