1.3 Statistical ensembles

An open system allows energy and mass transfer across its interface, a closed system allows transfer of energy only, but preventing mass transfer, and, finally, an

time, i.e., ρ ¼ ρð Þt . This balance is defined as the equation of continuity: ∂ρ ∂t

Transfer phenomenon of mass and energy are represented using the concept of flux, which is defined as a rate of passing a physical variable of interest across a unit cross-sectional area per unit time. If the flux is constant, input and output rates of a physical quantity within a finite volume are equal, and the density remains constant because a net accumulation within the systems is zero. If the flux varies spatially, specifically J ¼ Jð Þ x; y; z , then its density within the specified volume changes with

Many engineering processes occur in an open environment, having specific mass and energy transfer phenomena as practical goals. An exception is a batch reaction, where interfacial transport is blocked and a transient variation of the internal system is of concern. If the internal characteristic of the open system changes with time, the system moves toward a transient, non-equilibrium state. However, the transiency is subject to the human perception of the respective time scale. If engineering system performance is averaged over a macroscopic time scale, such as hours, days, and weeks, the time-averaged performance is a primary concern as those quantities can be compared with experimental data. Instead of transiency, the time to reach a steady state becomes more important in operating engineering processes because a steady-state operation is usually sought. Usually, the time to reach a steady state is much shorter than the standard operation time in a

In theoretical physics, statistical mechanics and fluid dynamics are not fully unified, and non-equilibrium thermodynamics is unsolved in theoretical physics. It is often assumed that the fluid flow is not highly turbulent, and a steady state is reached with a fully developed flow field. The thermodynamic characteristics are maintained within the steady flow, and the static equilibrium is assumed to be valid within small moving fluid elements. In such a situation, each fluid element can be

Nevertheless, a conflict between the thermodynamics and fluid dynamics stems from the absence of a clear boundary between the static equilibrium for isolated systems and the steady state of open systems. In principle, the steady state belongs to the non-equilibrium state although the partial differentials of any physical quantities are assumed to be zero (i.e., <sup>∂</sup>=∂<sup>t</sup> <sup>¼</sup> 0). A density does not change with time, but the flux exists as finite and constant in time and space. The time scale of particle motion can be expressed using the particle relaxation time defined as τ<sup>p</sup> ¼ m=β, where m and β exist as particle mass and Stokes' drag coefficient, respectively. The time scale for the fluid flow can be evaluated as the characteristic length divided by the mean flow speed, but the particle relaxation time scale is much shorter than the flow time scale. Therefore, the local equilibrium may be applied

In engineering, various dimensionless numbers are often used to characterize a system of interest. The Reynolds (Re) and Péclet (Pe) numbers indicate ratios of the convective transport to viscous momentum and diffusive heat/mass transport in a fluid, respectively. The Nusselt and Sherwood numbers represent ratios of the diffusion length scale as compared to the boundary layer thickness of the thermal

qualitatively analogous to a microstate of the thermodynamic ensemble.

without significant deviation from the real thermodynamic state.

þ ∇ � J ¼ 0 (1)

isolated system allows no transport across its interface.

Non-Equilibrium Particle Dynamics

steady state.

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1.2 Time scale and transiency

Thermodynamics often deals with macroscopic, measurable phenomena of systems of interest, consisting of objects (e.g., molecules or particles) within a volume. Statistical mechanics is considered as a probabilistic approach to study the microscopic aspects of thermodynamic systems using microstates and ensembles and to explain the macroscopic behavior of the respective systems.

Seven variables exist within statistical mechanics (i.e., temperature T, pressure P, and particle number N, which are conjugated to entropy S, volume V, chemical potential μ, and finally energy E of various forms). The thermodynamic ensemble uses the first and second laws of thermodynamics and provides constraints of having three out of the six variables (excluding E) remaining constant. The other three conjugate variables are theoretically calculated or experimentally measured. Statistical ensembles are either isothermal (for constant temperature) or adiabatic (of zero heat exchanged at interfaces). The adiabatic category includes NVE (microcanonical), NPH, μVL, and μPR ensembles, and isothermal ensembles possess NVT (canonical), NPT (isobaric-isothermal or Gibbs), and μVT (grand canonical). Here, ensembles of NVE and NPH are called microcanonical and isenthalpic, and those of NVT, μVT, and NPT are called canonical, grand canonical, and isothermalisobaric, respectively. Within statistical mechanical theories and simulations, canonical ensembles are most widely used, followed by grand canonical and isothermalisobaric ensembles. The adiabatic ensembles are equivalent to isentropic ensembles (of constant entropy) and are represented as NVS, NPS, μVS and μPS instead of NVE, NPH, μVL, and μPR, respectively. Non-isothermal ensembles often represent entropy S as a function of a specific energy form, of which details can be found elsewhere [1].
