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) - PowerPoint PPT PresentationTRANSCRIPT
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
time
position
entropy
Microscopic stochastic differential equations of motion (SDEs) for position and velocity.
SDE for entropy change: with positive mean production rate.
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
time
posi
tion
under forward protocol of driving
time
posi
tion
under reversed protocol
)(tx )(txR
Entropy change associated with a trajectory:
)(y trajectorseprob(rever
))(ctory prob(trajeln)]([tot tx
txktxs
R
0 tottot sS
In thermal equilibrium, for all trajectories 0 tot s
such thatSekimoto, Seifert, etc
Furthermore!
• trajectory entropy production may be split into three separate contributions – Esposito and van den Broek 2010, Spinney and Ford 2012
321tot ssss
0 1 s 0 2 s ? 3s
How to illustrate this?
• Non-stationary heat conduction
tem
per
atu
re
trap potential:force F(x) = -x
Trapped Brownian particle in a non-isothermal medium
position x
)(xTr
0
2
0 21)(
kT
xTxT T
r
0T
An analogy: an audience in the hot seats!
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)
)(xp
x
T
T
kT
xxp T
0
2
21)( q-gaussian
Steady heat current gives rise to entropy production. Now induce production.
2 s1 s
Steady heat current gives rise to entropy production. Now induce production.
2 s1 s
Particle explores space- and time-dependent background temperature:
Particle probability distribution
),( txp
x
warm wings
Particle probability distribution
),( txp
x
hot wings
Now the maths.....
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....
),(
),,(ln
),,(1
vxp
tvxp
t
tvxpdxdv
dt
sd
st
2
,2
),(
),(
),(
),,(
vxp
vxJ
txD
tvxpdxdv
dt
sd
st
irstv
Spinney and Ford, Phys Rev E 85, 051113 (2012)D
the remnant....
• only appears when there is a velocity variable • and when the stationary state is asymmetric in
velocity• and when there is relaxation
),(
),(ln
),,(3
vxp
vxp
t
tvxpdxdv
dt
sd
st
st
Simulations: distribution over position
Distribution over velocity at x=0 and various t
Approx mean total entropy production rate
spatial temperature gradientrate of change of temperature
1 s2 s
Mean ‘remnant’ entropy production is zero at this level of approximation
3 s
Comparison between average of total entropy production and the analytical approximation
Mean relaxational entropy production 1 s
Mean steady current-related entropy production 2 s
Distributions of entropy production ns
Some of the satisfy fluctuation relations!
)exp()(
)(tot
tot
tot ssp
sp
tots
)(/)(ln tottot spsp
ns
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
I SConclusions
• 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