# Temperature as the integral of kinetic energy?

Heat is the macroscopic phenomenon related to the kinetic energy of the particles. So, I wonder if there is a way to express the absolute temperature as the volume integral of the kinetic energy of the particles over a control volume.

I remember having found something like in Wikipedia, but without a proper reference, and I can't find it again anyway.

• In theory, you could just define temperature as $T=\oint_V v^2(\mathbf{x})\ dm$ – Quantumness May 17 '18 at 21:41
• That simple ? How do you manage the consistency of units between K and m²/s².kg ? – Aurélien Pierre May 19 '18 at 4:07

Yes there is. Temperature is related to the average energy per degree of freedom in a system. For a volume of ideal gas containing N molecules, each having three degrees of freedom, the energy integrated over the volume is $E = \frac{3}{2}Nk_BT$. $k_B$ is Boltmann's constant.

• By any chance, do you have the analytic form of the original integral that leads to this result ? – Aurélien Pierre May 19 '18 at 4:14
• All you need to know is the total number of degrees of freedom in the volume. In the ideal gas case this is 3 per molecule, so the integral is simply $\frac{3}{2}kT\int{dV\rho}$ where $\rho$ is the molecular number density. – my2cts May 19 '18 at 7:47

Given some particle distribution $f(t,\,\mathbf x,\,\mathbf v)$, one can define the pressure as, $$p=\frac{1}{3}\operatorname{Tr}\left[\int m\mathbf w\mathbf wf\left(t,\,\mathbf x,\,\mathbf v\right)\,\mathrm d^3x\right]$$ where $\mathbf w=\mathbf v-\mathbf V$ is the relative velocity where $\mathbf v$ is the particle velocity and $\mathbf V$ the bulk flow. We also define the particle density, $$n=\int f\left(t,\,\mathbf x,\,\mathbf v\right)\,\mathrm d^3x$$ Then, in thermodynamic equilibrium, the distribution function is essentially Maxwellian with some $T$, thus we can write $p=nT$ or, $$T=p/n=\frac{1}{3n}\operatorname{Tr}\left[\int m\mathbf w\mathbf wf\left(t,\,\mathbf x,\,\mathbf v\right)\,\mathrm d^3x\right]$$ which may or may not be convenient to use.

• I assume you write vectors in bold, but what are Tr and t here ? – Aurélien Pierre May 19 '18 at 4:04
• Tr is trace, t is time. – J.G. May 19 '18 at 6:10
• @AurélienPierre yes, vectors are bold, as is standard convention, $t$ is time and Tr is the trace. Probably could just do a scalar product of $\lVert\mathbf w\rVert$, but it's sometimes nice to be complete. – Kyle Kanos May 19 '18 at 10:08
• Standard just in America ;-) – Aurélien Pierre May 21 '18 at 23:24
• @AurélienPierre last I checked ISO 80000-2 standards are global, not American. – Kyle Kanos May 21 '18 at 23:26

I think what you're misremembering is an integral over temperature, such as $U=\int C_V dT$ expressing total thermal energy in terms of heat capacity.

• No, I remember this one. I'm really looking for an equivalence motion-temperature at molecular level over a control volume. Which seems reasonable since it's 2 different kinds of energy. – Aurélien Pierre May 19 '18 at 4:06