Table 1 : Historical milestones of governing equations for
multiphysics modeling
Within the foundations of statistical mechanics, any theory based on
bottom-up molecular models or top-down phenomenological models is
developed with the notion of microstates accessible by a system. The
dynamics of the system at this level can be described based on
transitions between microstates. A microstate defines the complete set
of configurations accessible to the system (e.g., positions and momenta
of all the particles/molecules of the system). For molecular systems
obeying laws of classical dynamics (Newton’s laws), the microstate of
the system with a given set of positions and momenta at a given time t
only depends on the microstate at the immediately preceding time step.
This memory-less feature is a hallmark of a Markov process, and all
Markov processes obey the master equation [5]. Note that the Markov
process is very general, and the classical dynamics is just a particular
case. The probability of access to a microstate defined by a given value
of the microstate variables y is denoted by P(y,t), which is
time-dependent for a general dynamical process at nonequilibrium. A set
of probability balance equations governs Markov processes (under certain
assumptions), collectively referred to as the master equation given by:
∂P(y,t)/∂t = ∫ dy’ [w(y | y’) P(y’,t) - w(y’|y)
P(y,t)]. (Eq. 1)
Here, y and y’ denote different microstates and w(y | y’) is
the transition probability (which is a rate of transition in units of a
frequency) from state y’ to state y.
The Liouville equation in classical dynamics is a particular case of the
continuous version of the master equation where the microstates are
enumerated by the positions and momenta of each particle [6, 7].
Newtonian dynamics obeys the Liouville equation and the parent master
equation, which is easy to see by recognizing that the elements of the
transition probabilities under Newtonian dynamics are delta functions
and therefore Newtonian dynamics trivially satisfies the Markov
property. Similarly, the Schrödinger equation, which governs the
dynamics of quantum systems is consistent with the quantum master
equation [8]. Therefore, the laws of classical and quantum dynamics
are both slaves to the master equation (Eq. 1). Neither the Schrödinger
equation nor Newton’s equations can predict the interactions between
systems (such as atoms and molecules), for which one needs to invoke
Maxwell’s equations to determine the nature of the potential energy
functions [9].
Macroscopic conservation equations can be derived by taking the
appropriate moment in (Eq. 1):
∂〈y〉/∂t = ∫∫ dy dy’ (y’ - y) w(y’ | y) P(y,t). (Eq. 2)
Here, 〈y〉 represents the average of y over all states, weighted by the
probability of accessing each state. Indeed, a particular case of the
master equation is the Boltzmann equation [10], where the
microstates defined in terms of the positions and momenta of all
particles are reduced to a one-particle (particle j) distribution by
integrating over the remaining n-1 particles. Here, the operator for the
total derivative d/dt is expressed as the operator for the partial
derivative ∂/∂t plus the convection term u • ∂/∂r, where u is the
velocity. The moments of the Boltzmann equation were derived by Enskog
for a general function yi (here i indexes the particle)
[10]. Substituting y as mi, the mass of particle i
yields the continuity equation, as mivi,
the momentum of particle i yields the momentum components of the
Navier-Stokes equation, and as 1/2
mivi2, the kinetic
energy of the particle, yields the energy equation, which together
represents conservation equations that are the pillars of continuum
hydrodynamics. Similarly, the rate equations for describing the
evolution of species concentrations of chemical reactions can be
obtained by computing the moment of the number of molecules using an
analogous version of (Eq. 2) known as the chemical master equation
[5].