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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)

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entropy production due to non stationary heat conduction

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
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
Stochastic thermodynamics
  • (Arguably) the best available representation of irreversibility and entropy production
slide4

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
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
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 trajectory1
Entropy change associated with a trajectory:

Sekimoto, Seifert, etc

such that

In thermal equilibrium, for all trajectories

furthermore
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
How to illustrate this?
  • Non-stationary heat conduction
slide10

Trapped Brownian particle in a non-isothermal medium

trap potential:

force F(x) = -x

temperature

position x

slide14
Stationary distribution of a particle in a harmonic potential well () with a harmonic temperature profile (T)

q-gaussian

n b this probability distribution is a variational solution to kramers equation
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.
and some more maths
and some more maths....

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

D

the remnant
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
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

where are we now
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

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
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