# Statistical averages - integration in phase space

I apologize for my horrible math skills. But how exactly is one supposed to get ensemble averages in the 6N dimensional phase space?

Suppose I have two particles in 1D. The phase space element is then often written as $$d\Gamma = dx_1dx_2dp_1dp_2$$.

For a physical variable $$A(q,p)$$ (such as $$xp$$, $$x^2$$), do we calculate the 'ensemble' average this way?

$$(t) = \int^\infty_{-\infty} \int^\infty_{-\infty}\int^\infty_{-\infty} \int^\infty_{-\infty}A(x,p)\rho(x,x,p,p,t)dxdxdpdp$$ where please notice I am omitting all indexes and have just generic $$x$$ and $$p$$.

Am I doing it wrong? I am confused and not sure whether I should be somehow keeping the indexes. And also, should I be dividing by $$2$$ because the density is normalized to $$N$$?

Let's work in 1D with two particles. The phase space element is $$d\Gamma = dx_1dx_2dp_1dp_2$$.

The probability density in the phase space of two particles is a function $$\rho(x_1,x_2,p_1,p_2)$$. When multiplied by $$d\Gamma$$, it will be the probability of finding position of particle 1 close to $$x_1$$, position of particle 2 close to $$x_2$$, momentum of particle $$1$$ close to $$p_1$$ and similarly for $$p_2$$. One can see from the description that all the four arguments matter for their name/meaning as well as for their values. And this is important not only when dealing with probabilities, but also for evaluating averages.

When one is dealing with a physical observable $$A(q,p)$$ (such as $$xp$$, $$x^2$$), what exactly do we mean?

$$ = \int^\infty_{-\infty} \int^\infty_{-\infty}\int^\infty_{-\infty} \int^\infty_{-\infty}A(x,p)\rho(x,x,p,p)dxdxdpdp$$

is mathematically meaningless.

First of all we have to understand what is the observable we want to average.

We may have observables related to a specific particle or collective observable.

Let's see an example for each case:

1. $$A_1 = x_1$$. $$\left< A_1 \right>$$ represents the average value of the coordinate of particle 1: $$ = \int\int\int\int x_1\rho(x_1,x_2,p_1,p_2) d\Gamma$$ where the integral is extended over the relevant part of the phase space (it depends on the ensemble and on the physical system.
2. $$A_2 = \sum_{i=1}^2\frac{p_i^2}{2m}$$. $$\left< A_2 \right>$$ represents the average value of the kinetic energy of the system of two particles: $$ = \int\int\int\int \left(\frac{p_1^2+p_2^2}{2m}\right)\rho(x_1,x_2,p_1,p_2) d\Gamma$$.

Notice, that, if $$\rho$$ is properly normalized, no additional factors are required.