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

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


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? (SDEs) for position and velocity.

  • 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 (SDEs) for position and velocity.

  • 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: (SDEs) for position and velocity.

Sekimoto, Seifert, etc

such that

In thermal equilibrium, for all trajectories


Furthermore
Furthermore! (SDEs) for position and velocity.

  • 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? (SDEs) for position and velocity.

  • Non-stationary heat conduction


Trapped Brownian particle in a (SDEs) for position and velocity.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! (SDEs) for position and velocity.


An analogy an audience in the hot seats1
An analogy: an audience in the hot seats! (SDEs) for position and velocity.

steady mean heat conduction


An analogy an audience in the hot seats2
An analogy: an audience in the hot seats! (SDEs) for position and velocity.

steady mean heat conduction


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

q-gaussian





Particle probability distribution
Particle probability distribution temperature:

warm wings


Particle probability distribution1
Particle probability distribution temperature:

hot wings


Now the maths
Now the maths..... temperature:


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.... to Kramers equation

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

D


The remnant
the remnant.... to Kramers equation

  • 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 to Kramers equation over position


Distribution over velocity at x 0 and various t
Distribution over velocity at to Kramers equation x=0and various t


Approx mean total entropy production rate
Approx mean total entropy production rate to Kramers equation

spatial temperature gradient

rate of change of temperature

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



Mean relaxational entropy production
Mean relaxational entropy production the analytical approximation



Distributions of entropy production
Distributions of entropy production the analytical approximation


Some of the satisfy fluctuation relations
Some of the satisfy fluctuation relations! the analytical approximation


Where are we now
Where are we now? the analytical 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


Conclusions

I S the analytical approximation

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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