Contents - Max-Planck-Institut für Physik komplexer Systeme

2.14 Work Dissipation Along a Non Quasi-Static Process
LÉO GRANGER, MARKUS NIEMANN, HOLGER KANTZ
Many complex dynamical phenomena which have traditionally
been interpreted in the framework of dynamical
systems have more recently been considered
from a statistical physics perspective. Most dynamical
models, among them models for lasers, chemical
reactions, or population dynamics, describe open systems,
i.e., systems which are subject to some throughput
of energy. In a more statistical setting, one would
couple such models to one or more heat baths rather
than relying on a perfectly deterministic description.
Such systems are thermodynamic systems out of equilibrium,
and their fluctuations, both of dynamical origin
and due to the heat exchange with the baths, are
non-equilibrium fluctuations. In the past two decades,
much theoretical progress has been made towards a
characterization of such fluctuations by fluctuation theorems.
Even though these theorems do not yet cover
the physically most relevant and interesting scenarios,
they make important statements.
In classical thermodynamics, the maximum work theorem
states that the maximum work −Wr 1 that can be extracted
when driving a system between two given equilibrium
states is obtained for a reversible process [1].
Such a process does not produce any entropy. If the
system is in contact with a heat bath at constant temperature
T , then a process is reversible if and only if
it is quasi-static. In this case the reversible work Wr is
the difference in free energy ∆F between the final and
the initial state. If the system is quasi-statically driven
from the final state back to the initial state, then the
same amount of work Wr = ∆F will be retrieved from
the system. However, if the system is driven at a finite
speed, then the work performed during the process
is random and is generally greater than the reversible
work. The excess work Wd = W − ∆F is dissipated
to the reservoir in the form of heat, leading to a total
entropy production ∆iS = Wd/T . Crooks’ fluctuation
relation is a statement about the asymmetry of the work
distributions during a non quasi-static process and the
corresponding reverse process. It links the probability
P(W) to perform a certain amount W of work when
driving the system from the initial state to the final state
to the probability ¯ P(−W) of performing the opposite
amount when performing the reverse process [2]:
P(W) W − ∆F
¯P(−W)
= exp . (1)
kBT
The fluctuating lattice-Boltzmann model (FLBM) is a
stochastic lattice model for a thermal ideal gas [3]. It
provides a simple model for investigating the fluctuating
dynamics of isothermal processes [4]. The FLBM
simulates an ideal gas in contact with a heat bath and
subjected to a force field per unit mass f = − ∇φλ, derived
from a potential φλ(r). It consists of mass densities
{ni} b i=1 moving along the edges of a Bravais lattice
{r} according to a finite set of b velocities {ci} b i=1 .
The dynamics takes place in discrete time. Each time
step is divided into a collision and a propagation step.
During the propagation step the post-collisional populations
{n∗ i } are simply propagated according to the set
of velocities {ci}:
ni(r + ci,t + 1) = n ∗ i (r,t). (2)
During the collision step the populations at each node
are randomly shuffled such that at each node r the
mass ρ(r,t) =
i ni(r,t) and momentum j(r,t) =
i ni(r,t)ci densities are exactly conserved and that
the local stress
i nici,αci,β relaxes to and fluctuates
around the Euler stress ρkBTδα,β +ρvαvβ, where α and
β are the Cartesian coordinates, δα,β = 1 if α = β and 0
otherwise, and v = j/ρ is the local fluid velocity 2 . The
effect of the body force f is to increase the momentum
density by ρ f at each time step.
At thermal equilibrium, the density at point r fluctuates
around its mean value given by the Boltzmann factor
〈ρ(r)〉eq ∝ exp(−φλ(r)/kBT). The potential energy
V [ρ,φλ] =
r ρ(r)φλ(r) of the system changes during
one time step as λ changes to λ + δλ, with an amount
of work δW performed on the system:
δW = V [ρ,φλ+δλ] − V [ρ,φλ] =
r
ρ(r) ∂φλ
δλ (3)
∂λ
for small δλ. The total work W performed when
switching λ from 0 to 1 in steps of δλ is given by:
W =
1
0
δW
δλ
dλ. (4)
The time reversed process is obtained by switching λ
from 1 back to 0 in steps of δλ.
Fig. 1 shows examples of distributions of the work performed
during a non quasi-static process (solid lines)
and the work extracted during the corresponding reverse
process (dashed lines). These distributions were
obtained with the potential
φλ(r) = λA cos 2π x
+ 1
(5)
l
where A = 0.01 is the amplitude of the potential and
l = 100 the length of the lattice.
1 We consider the work performed on the system, which is the opposite of the work extracted.
2 For a detailed description of the dynamics of the model, see [3]
68 Selection of Research Results