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.