2.1 Thermal and Brownian effects
One of the main attributes of statistical mechanics of equilibrium and
nonequilibrium systems that differentiate it from traditional
hydrodynamics is that the kinematics and thermal effects have to be
treated with equal importance. It is worth noting that while the thermal
effects and fluctuations are described within the scope of the master
equation (Eq. 1), by taking the moment (average) to derive the
conservation law (Eq. 2), often the thermal effects are averaged out to
produce only a mean-field equation. Indeed, the continuity, momentum
(Navier-Stokes), and energy equations cannot accommodate thermal
fluctuations that are inherent in Brownian motion even though such
effects are fully accommodated at the level of the parent master
equation. Therefore, nanoscale fluid dynamics (NFD) must be approached
differently than traditional hydrodynamics.
One approach starts with the mean-field conservation equation, such as
the Boltzmann equation, and adds the thermal fluctuations as a random
forcing term, which results in the Boltzmann-Langevin equation derived
by Bixon and Zwanzig [11]. This approach amounts to random
fluctuating terms being added as random stress terms to the
Navier-Stokes equations. The above procedure, referred to as the
fluctuating hydrodynamics (FHD) approach, was first proposed by Landau
and Lifshitz [12]. In the FHD formulation, the fluid domain
satisfies:
∇ • u = 0
ρDu/dt = ρ[∂u/∂t + u • ∇u] = ∇ • σ, (Eq. 3)
where, u and ρ are the velocity and density of the fluid respectively,
and σ is the stress tensor given by, σ = pJ + µ [∇u +
(∇u)T] + S. Here, p is the pressure, J is the
identity tensor, and µ is the dynamic viscosity. The random stress
tensor S is assumed to be a Gaussian white noise that satisfies:
〈Sij(x,t)〉=0
〈Sij(x,t) Slm(x’,t’)〉 =
2kBTµ (δil δkm +
δim δkl) δ(x-x’) δ(t-t’), (Eq. 4)
where, 〈•〉 denotes an ensemble average, kBT is the
Boltzmann constant, T is the absolute temperature, and
δij is the Kronecker delta. The Dirac delta functions
δ(x-x’) and δ(t-t’) denote that the components of the random stress
tensor are spatially and temporally uncorrelated. The mean and variance
of the random stress tensor of the fluid are chosen to be consistent
with the fluctuation-dissipation theorem [13]. By including this
stochastic stress tensor due to the thermal fluctuations in the
governing equations, the macroscopic hydrodynamic theory is generalized
to include the relevant physics of the mesoscopic scales ranging from
tens of nanometers to a few microns.
An alternative approach to NFD (and one that is different from FHD) is
to start with a form of the master equation referred to as the
Fokker-Planck equation. Formally, the Fokker-Planck equation is derived
from the master equation by expanding w(y’ | y) P(y,t) as a
Taylor series in powers of r=y’-y. The infinite series is referred to as
the Kramers-Moyal expansion, while the series truncated up to the second
derivative term is known as the Fokker-Planck or the diffusion equation,
which is given by [5]:
∂P(y,t)/∂t = - ∂/∂y [a1(y)P] +
∂2/∂y2 [a2(y)P].
(Eq. 5)
Here, an(y)=∫ rn w(r) dr. The solution
to the Fokker-Planck equation yields the probability distributions of
particles which contain the information on Brownian effects. At
equilibrium (i.e., when all the time-dependence vanishes), the solution
can be required to conform to the solutions from equilibrium statistical
mechanics. This approach leads to a class of identities for transport
coefficients, including the famous Stokes-Einstein diffusivity for
particles undergoing Brownian motion to be discussed later in this
article. Furthermore, there is a one-to-one correspondence between the
Fokker-Planck equation and a stochastic differential equation (SDE) that
describes the trajectory of a Brownian particle. The generalized
Fokker-Planck equation is written in terms of a generalized order
parameter (or sometimes referred to as a collective variable) S, given
by:
∂P(S,t)/∂t = [D/kBT] ∂/∂S [P(S,t) ∂F(S)/∂S] + D
∂2P(S,t)/∂S2, (Eq. 6)
where, F(S) is the free energy density (also referred to as the Landau
free energy) along S [14], D is the diffusion coefficient along S,
which is also related to the an’s of the original
Fokker-Planck equation, i.e., a2=2D. The quantity
kBT, which has the units of energy, is called the
Boltzmann factor and serves as a scale factor for normalizing energy
values in NFD. Corresponding to every generalized Fokker-Planck equation
(Eq. 6), there exists a SDE given by:
∂S/∂t = - [D/kBT] ∂F(S)/∂S +
(2D)1/2 ξ(t), (Eq. 7)
where, ξ(t) represents a unit-normalized white noise process. The SDE
encodes for the Brownian dynamics (BD) of the particle in the limit of
zero inertia. When the inertia of the particle is added, the
corresponding equation is often referred to as the Langevin equation
[13]. In summary, Brownian or thermal effects are described in the
hydrodynamics framework, either using the FHD or the BD/Langevin
equation approach.