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As part of the 2019 revision of the SI, the Boltzmann constant is one of the seven "defining constants" that have been defined so as to have exact finite decimal values in SI units. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly 1.380 649 × 10 −23 joules per kelvin. [1]
The general equation can then be written as [6] = + + (),. where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions.
Observing the previous equation, a trivial solution is found for the case dc/dξ = 0, that is when concentration is constant over ξ.This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the ...
kT (also written as k B T) is the product of the Boltzmann constant, k (or k B), and the temperature, T.This product is used in physics as a scale factor for energy values in molecular-scale systems (sometimes it is used as a unit of energy), as the rates and frequencies of many processes and phenomena depend not on their energy alone, but on the ratio of that energy and kT, that is, on E ...
A different interpretation of the lattice Boltzmann equation is that of a discrete-velocity Boltzmann equation. The numerical methods of solution of the system of partial differential equations then give rise to a discrete map, which can be interpreted as the propagation and collision of fictitious particles.
The Boltzmann equation is notoriously difficult to integrate. David Hilbert spent years trying to solve it without any real success. The form of the collision term assumed by Boltzmann was approximate. However, for an ideal gas the standard Chapman–Enskog solution of the Boltzmann equation is
Chapman–Enskog theory provides a framework in which equations of hydrodynamics for a gas can be derived from the Boltzmann equation. The technique justifies the otherwise phenomenological constitutive relations appearing in hydrodynamical descriptions such as the Navier–Stokes equations .
In statistical thermodynamics, thermodynamic beta, also known as coldness, [1] is the reciprocal of the thermodynamic temperature of a system: = (where T is the temperature and k B is Boltzmann constant).