Classical approximation

Before discussing a possible approximation for many-body systems we will first study the classical limit of inverse quantum statistics. The classical limit is much easier to solve than the full quantum mechanical problem and may, for example for large masses, already give a useful approximation.

The phase space density of a classical canonical ensemble is given by

$$p(x,p_{\rm cl}\vert v)= Z^{-1} e^{-\beta\left(\frac{p_{\rm cl}^2}{2m}+v(x)\right)} ,$$ (55)

with

$$Z = \int\! dp_{\rm cl}\,dx\, e^{-\beta\left(\frac{p_{\rm cl}^2}{2m}+v(x)\right)} .$$ (56)

Here we used $p_{\rm cl}$ to denote the classical momentum to distinguish it from a density $p$. The probability $p(x\vert v)$ for measuring $x$ [to simplify the notation we abstain in this context from denoting the observable $O$ explicitly] is then obtained by integrating over $p_{\rm cl}$,

$$p(x\vert v) = \int\! dp_{\rm cl}\, p(x,p_{\rm cl}\vert v) = Z_x^{-1}e^{-\beta v(x)} ,$$ (57)

where

$$Z_x= \int\! dx\,e^{-\beta v(x)} .$$ (58)

Notice, that the classical $p(x\vert v)$ is mass independent, and, most important, that it can be obtained directly from $v(x)$ without having to solve an eigenvalue problem like in the quantum case.

Analogously to the quantum mechanical approach the classical likelihood model (57) for position measurements can now be combined with a prior model for potentials $v$, leading to a posterior density $p(v\vert D)$. In particular, adding a Gaussian process prior the log-posterior becomes

$$\ln p(v\vert D) = -\beta \sum_{i=1}^n v(x_i) -\frac{\lambda}... ...det\left( \frac{\lambda}{2\pi}{\bf K}_0\right)^{\frac{1}{2}} .$$ (59)

Again, we may refer to a maximum posterior approximation and consider the potential which maximizes the posterior as the solution of our reconstruction problem. The corresponding stationarity equation is found by setting the functional derivative of the log-posterior with respect to $v(x)$ to zero,

$$0 = \delta_{v}\ln p(v\vert D) = -\beta N-\lambda{\bf K}_0 (v-v_0) + n \beta p(x\vert v) .$$ (60)

Choosing an initial guess $v^{(0)}$ Eq. (60) can be solved by straightforward iteration. The results of a classical calculation (with parameters and data as in Fig. 4, but without energy penalty term) are shown in Fig. 8.

Figure 8: Classical approximation of symmetric potential. Shown are likelihoods (left hand side) and potentials (right hand side): Original likelihood and potential (thin lines), approximated likelihood and potential $v_{\rm cl}$ (thick lines), empirical density (bars). The dotted line shows $v_{\rm cl}-c$ with constant $c$ = ${\rm min}[v_{\rm cl}]-{\rm min}[v_{\rm true}]$. Except for the fact that no energy penalty term has been used for this classical calculation the parameters and data are the same as in Fig. 4. (20 data points, sampled from the true quantum mechanical likelihood, truncated RBF covariances (54) with $\sigma_{\rm RBF}$ = $7$, $\lambda$ = $0.001$, inverse physical temperature $\beta$ = 1, $v(x)$ = $v(-x)$ and $v$ = 0 at the boundaries.)