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
