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].