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)

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Entropy production due to non-stationary heat conduction

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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


Entropy production due to non stationary heat conduction

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


Entropy production due to 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

An analogy: an audience in the hot seats!


An analogy an audience in the hot seats1

An analogy: an audience in the hot seats!

steady mean heat conduction


An analogy an audience in the hot seats2

An analogy: an audience in the hot seats!

steady mean heat conduction


Entropy production due to non stationary 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.


Steady heat current gives rise to entropy production now induce production1

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


Particle explores space and time dependent background temperature

Particle explores space- and time-dependent background temperature:


Particle probability distribution

Particle probability distribution

warm wings


Particle probability distribution1

Particle probability distribution

hot wings


Now the maths

Now the maths.....


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


Simulations distribution over position

Simulations: distribution over position


Distribution over velocity at x 0 and various t

Distribution over velocity at x=0and various t


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


Comparison between average of total entropy production and the analytical approximation

Comparison between average of total entropy production and the analytical approximation


Mean relaxational entropy production

Mean relaxational entropy production


Mean steady current related entropy production

Mean steady current-related entropy production


Distributions of entropy production

Distributions of entropy production


Some of the satisfy fluctuation relations

Some of the satisfy fluctuation relations!


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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