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If $\rho$ is the mass density of a fluid and $A({\bf v})$ is an function of the velocity, which is distributed according to $f({\bf v})$, we have an averaging process

$A\mapsto \langle A\rangle:=\int A({\bf v})\ f({\bf v})\ \mathrm d^3v.$

We now might define the pressure and energy in the frame of the flow as

$P_{ij}:=\rho\ \langle (v_i-\langle v_i \rangle)(v_j-\langle v_j \rangle) \rangle,$

$E_\text{Kin}:=\rho\ \langle \tfrac{1}{2} ({\bf v}-\langle {\bf v} \rangle)^2 \rangle.$

I realize that $$P=\rho\ \text{Cov}(v_i,v_j),$$ $$E_\text{Kin}=\tfrac{1}{2}\sum_i \text{Var}(v_i).$$ So they are given by covariance (see also covariance matrix) and variance for $f$.

Can I formulate the heat flow

$Q_i:=\rho\cdot \langle \tfrac{1}{2} ({\bf v}-\langle {\bf v} \rangle)^2 (v_i-\langle v_i \rangle) \rangle$

also in terms of general statistical functionals of ${\bf v}$? E.g. some functional of skewness?

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  • $\begingroup$ Your question is a little vague, but in brief, I don't think you can formulate heat flow that way. Heat flow is a transport phenomenon, not solely a property of the local statistical distribution of velocity. Heat flow is nothing but transport of energy. $\endgroup$ – SimpleLikeAnEgg Nov 12 '13 at 19:09
  • $\begingroup$ @SimpleLikeAnEgg: Given as an expectation value like above, and in comparison with the other quantities, which has such an interpretation, why wouldn't it be a property of the statistical distibution. It's just third order in $\bf v$. $\endgroup$ – Nikolaj-K Nov 12 '13 at 20:45
  • $\begingroup$ Yes, on second thought, what you have defined there is a property and type of "heat flow," but not a complete one in the sense of usefulness. In terms of kinetic theory of gases, it would correspond to "collisionless" diffusion of molecules. Perhaps in some situations that is useful, but for many of interest defining a meaningful heat flow quantity requires further consideration of collisions. $\endgroup$ – SimpleLikeAnEgg Nov 12 '13 at 21:34
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Heat flow is nothing but energy transport. Energy transport is achieved in a number of ways. What you have defined with $$Q_i:=\rho\cdot \langle \tfrac{1}{2} ({\bf v}-\langle {\bf v} \rangle)^2 (v_i-\langle v_i \rangle) \rangle$$ is simply one term that could be expected to relate to heat flow rate. In fact, the $Q_i$ you have defined corresponds to the instantaneous heat flow rate from mass diffusion for a non-interacting medium. A more realistic consideration of heat flow in a medium would involve a model for interactions.

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