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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

- 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

- (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

- 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

- the relative likelihood of observing reversed behaviour

position

position

time

time

under forward protocol of driving

under reversed protocol

Sekimoto, Seifert, etc

such that

In thermal equilibrium, for all trajectories

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

- Non-stationary heat conduction

Trapped Brownian particle in a non-isothermal medium

trap potential:

force F(x) = -x

temperature

position x

steady mean heat conduction

steady mean heat conduction

q-gaussian

warm wings

hot wings

- distribution valid in a nearly-overdamped regime
- maximisation of the Onsager dissipation functional
- which is related to the entropy production rate.

Spinney and Ford, Phys Rev E 85, 051113 (2012)

D

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

spatial temperature gradient

rate of change of temperature

Mean ‘remnant’ entropy production is zero at this level of approximation

- 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

I S

- 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