Boltzman’s equation

While no one debates the validity of Boltzman s equation (equation 9.32) - one good reason for this is that there has been no experimental evidence suggesting that the theory is wrong - it is in a very important sense laced with a fundamental... 

For more general interactions - i.e. not necessarily between two hard-spheres - we introduce a differential scattering cross section a v,8), defined by b db d(j> a u,8)dU. Boltzman s equation (equation 9.32) becomes... 

The general problem of finding nonequilibrium solutions to Boltzman s equation is, as already mentioned above, an exceedingly difficult problem. Two tools that have proven invaluable in providing insight into nonequilibrium phenomena,... 

We can obtain a more immediately useful form of the conservation theorem by directly multiplying Boltzman s equation (equation 9.40) by K and integrating over V. Noting that the RHS vanishes identically due to the general conservation theorem we have just proven, we have... 

Euler s equation (equation 9.7) may be recovered from Boltzman s equation as a consequence of the conservation of momentum, but only in the zeroth-order approximation to the full distribution function. Setting k — mvi in equation 9.52 gives, in component form. 

Using the first condition in Boltzman s equation (equation 9.80), we have... 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Boltzman s constant and T is the temperature. Using equations 7.101 and 7.102, we see that this is equivalent to the following PCA rule ... 

Assuming fj, < 1/2, this solution implies a monotonic approach to equilibrium with time. From a purely statistical point of view, this is certainly correct the difference in number between the two different balls decreases exponentially toward a state in which neither color is preferred. In this sense, the solution is consistent with the spirit of Boltzman s H-theorem, expressing as it does the idea of motion towards disorder. But the equation is also very clearly wrong. It is wrong because it is obviously inconsistent with the fundamental properties of the system it violates both the system s reversibility and periodicity. While we know that the system eventually returns to its initial state, for example, this possibility is precluded by equation 8.142. As we now show, the problem rests with equation 8.141, which must be given a statistical interpretation. 

The fact that the condition dll/dl = 0 is the same as the condition of detailed-balance, and therefore that equation 9.34 is a necessary condition for the solution of equation 9.33, follows from the proof of Boltzman s H-Theorem ... 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

In order to get this expression into a more familiar form (equation 9.7), we now consider the zeroth-order approximation to /. We assume that / is locally a Maxwell-Boltzman distribution, and treat the density p, temperature T[x,t) = < V — u p> (where k is Boltzman s constant), and average velocity u all as slowly changing variables with respect to x and t. We can then write... 

Euler s equation is thus recovered as a direct consequence of momentum conservation, but only via the zeroth-order approximation to the full solution to the Boltzman-equation. 

In classical molecular dynamics simulations, atoms are generally considered to be points which interact with other atoms by some predehned potential form. The forms of the potential can be, for example, Lennard-Jones potentials or Coulomb potentials. The atoms are given velocities in random directions with magnitudes selected from a Maxwell-Boltzman distribution, and then they are allowed to propagate via Newton s equations of motion according to a finite-difference approximation. See the following references for much more detailed discussions Allen and Tildesley (1987) and Frenkel... 

We start with the simpler model, for the electron gas. In a metal, the electron density is so high that the Pauli exclusion principle must be taken into account, i.e., there can be only one electron in each quantum state. The result is that the electrons obey the Fermi distribution rather than Boltzman s, and may be considered strongly degenerate (8). The equation of state is... 

Assumption of Stokes flow to describe the resin flow between the tows and Brinkman s equation to describe the flow inside the tows (in this case, it is useful to use the Lattice-Boltzman method as a numerical approach [80]). 

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