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Entropy production due to non-stationary heat conduction

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### 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

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

Stochastic thermodynamics

- (Arguably) the best available representation of irreversibility and entropy production

Microscopic stochastic differential equations of motion (SDEs) for position and velocity.

SDE for entropy change: with positive mean production rate.

entropy

position

time

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

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

Entropy change associated with a trajectory:

Sekimoto, Seifert, etc

such that

In thermal equilibrium, for all trajectories

Furthermore!

- trajectory entropy production may be split into three separate contributions
- Esposito and van den Broek 2010, Spinney and Ford 2012

How to illustrate this?

- Non-stationary heat conduction

Trapped Brownian particle in a non-isothermal medium

trap potential:

force F(x) = -x

temperature

position x

An analogy: an audience in the hot seats!

steady mean heat conduction

An analogy: an audience in the hot seats!

steady mean heat conduction

Stationary distribution of a particle in a harmonic potential well () with a harmonic temperature profile (T)

q-gaussian

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

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

Particle probability distribution

warm wings

Particle probability distribution

hot wings

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.

the remnant....

- only appears when there is a velocity variable
- and when the stationary state is asymmetric in velocity
- and when there is relaxation

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

Comparison between average of total entropy production and the analytical approximation

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

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

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