PHYSICAL REVIEW E 85, 041117 (2012)
Study of nonequilibrium work distributions from a fluctuating lattice Boltzmann model
S. Siva Nasarayya Chari,
1
K. P. N. Murthy,
1
and Ramarao Inguva
2
1
School of Physics, University of Hyderabad, Hyderabad 500046, India
2
Visiting Professor, Centre for Modelling, Simulation and Design, University of Hyderabad, Hyderabad 500046, India
(Received 29 December 2011; published 12 April 2012)
A system of ideal gas is switched from an initial equilibrium state to a final state not necessarily in equilibrium,
by varying a macroscopic control variable according to a well-defined protocol. The distribution of work
performed during the switching process is obtained. The equilibrium free energy difference, F , is determined
from the work fluctuation relation. Some of the work values in the ensemble shall be less than F . We term
these as ones that “violate” the second law of thermodynamics. A fluctuating lattice Boltzmann model has
been employed to carry out the simulation of the switching experiment. Our results show that the probability
of violation of the second law increases with the increase of switching time (τ ) and tends to one-half in the
reversible limit of τ →∞.
DOI: 10.1103/PhysRevE.85.041117 PACS number(s): 05.70.Ln, 05.40.−a, 47.11.Qr
I. INTRODUCTION
Consider a system driven isothermally from an equilibrium
state A to a state B , not necessarily in equilibrium. The second
law of thermodynamics can be stated in terms of the work W
performed during the process and the equilibrium free energy
difference F (= F
B
− F
A
) as
W F. (1)
In the above, equality obtains when the process is reversible.
In the language of statistical mechanics, we should write the
second law as
〈w〉 F. (2)
Here w, the work performed, is a random variable and 〈· · ·〉
implies averaging over an ensemble of work values. The
fraction of work values in the ensemble which are less than
F , is usually termed as the probability of “violation” of
second law, denoted by p(τ ). Here τ denotes the switching
time, defined as the time taken to drive the system from A to
B . Without loss of generality we assume the switching to be
uniform over time τ . Formally we write
p(τ ) =
F
−∞
ρ (w,τ )dw, (3)
where ρ (w, τ ) is distribution of work, for a given switching
time τ . Our aim in this paper is to calculate p(τ ) as a function
of τ for a uniform switching protocol. We have employed the
fluctuating lattice Boltzmann model (FLBM) [1–3] to simulate
the switching experiment on a system of ideal gas.
In the next section, we present a brief description of FLBM
followed by details of the switching experiment and results.
We observe from the results, that p(τ ) increases with τ and
tends to one-half in the reversible limit (τ →∞). We explain
this counter intuitive result through an analytical argument
from Jarzynski equality.
II. FLUCTUATING LATTICE BOLTZMANN MODEL
The lattice Boltzmann methods [4–6] have emerged as
a powerful computational tool in the study of complex
phenomena in fluid flows and in hydrodynamics, particularly
in complex geometries. Fluctuating lattice Boltzmann model
had been devised for soft matter systems, where thermal
fluctuations cannot be neglected.
In this work, we consider a two-dimensional square lattice
with nine discrete velocities :{ c
i
,i = 0, ··· ,8} at each lattice
node, see Fig. 1. Lattice constant b and the time step h are taken
to be unity. Mass of a single particle m
p
per unit volume is
taken as the fluctuation parameter, μ = m
p
/b
2
. The number
of particles, N
p
at each lattice node is given by
N
p
=
ρb
2
m
p
=
ρ
μ
. (4)
From the above, the equation of state of an ideal gas can be
written as
k
B
T = μc
2
s
, (5)
where
c
s
=
∂p
∂ρ
is the isothermal speed of sound and k
B
is Boltzmann’s con-
stant. Since the parameter μ is directly related to temperature,
it controls the amount of thermal fluctuations in the system.
Since, for an ideal gas, N
p
is Poisson distributed, the relative
importance of fluctuations is quantified by
Bo =
(
〈N
2
p
〉−〈N
p
〉
2
)
1/2
〈N
p
〉
=
1
〈N
p
〉
1/2
=
μ
ρ
1/2
, (6)
where Bo is called as the Boltzmann number [3]. Let n
i
( r,t )
denote the density of ideal gas molecules along the velocity
direction c
i
at lattice location r , at time t . The hydrodynamic
variables, mass density ρ ( r,t ) and momentum density
j ( r,t )
are obtained as
ρ ( r,t ) =
8
i =0
n
i
( r,t ), (7)
j ( r,t ) =
8
i =0
n
i
( r,t ) c
i
. (8)
041117-1 1539-3755/2012/85(4)/041117(6) ©2012 American Physical Society