# Mathematical modeling technique of entropy increasing and decreasing process

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The entropy(both thermodynamic and computational) increasing and decreasing(e.g. caused by the manipulation of some Maxwell's demon) of a system, e.g. the formation or decomposition of structures, sometimes is a gradual process. Is there a mathematical modeling technique in some field of theoretical physics can be applied to measuring or monitoring such a process?

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You can use Boltzmann's H-Theorem to compute the entropy increase or decrease with time.

Consider you have a collision term $C(f)$ for the probability Distribution function $f$, where it holds e.g. $\frac{Df}{Dt} = C(f)$. Let $s = \ln f$ (generates the entropy density); multiply the collision term by $s$ and integrate over the phase space $\Sigma$. Then you have (<> denotes the average)

$\frac{D<s>}{Dt} = \int_{\Sigma}d \sigma sC(f).$ (*)

Suppose that there exists an equilibrium probability Density $f_0$ with $C(f_0) = 0$ (no entropy production). The Distribution function depends on all of the the $i$-th Phase space variables $x_i$. From this you can expand the non-equilibrium probability Density in Terms of the Equilibrium Density by the series expansion

$f = (1+ (<x_i> - <x_i>_0)\frac{\partial}{\partial x_i} + (<x_ix_j>-<x_j><x_i>_0-<x_j>_0<x_i>$

$-<x_ix_j>_0)\frac{\partial^2}{\partial x_i \partial x_j} + \dots)f_0$ (Summation convention is used).

The $<>_0$ is averaging with Equilibrium Distribution function; These are also known. You can convince yourself, that this Expansion holds by taking various Moments of this and using that Integration of a total derivative vanishes.

Substituting this Expansion into the equation (*) gives you an Expansion of the entropy production rate in all nonequilibrium Moments $<x_i>,<x_ix_j>$. You will have a Connection between the entropy Change and some other quantities that are easy to measure.

answered Oct 19, 2018 by (40 points)

@PatrickLinker Would you please recommend me any materials(books, papers, etc.) elaborating Boltzmann's H-Theorem available from Internet free or not free, please?

Thanks a lot!

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