Statistical Mechanics · Problem 6.4

Ideal Bose Systems — R. K. Pathria

Physics — Statistical Mechanics · charged-particle-equilibrium-density

Problem

The potential energy of a system of charged particles, characterized by particle charge ee and number density n(r)n(\mathbf{r}), is given by U=e22∫n(r)n(r′)∣r−r′∣ dr dr′+e∫n(r)ϕext(r) dr,U = \frac{e^2}{2}\int\frac{n(\mathbf{r})n(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|}\,d\mathbf{r}\,d\mathbf{r}' + e\int n(\mathbf{r})\phi_{\text{ext}}(\mathbf{r})\,d\mathbf{r}, where ϕext(r)\phi_{\text{ext}}(\mathbf{r}) is the potential of an external electric field. Assume that the entropy of the system, apart from an additive constant, is given by the formula S=−k∫n(r)ln⁡n(r) dr;S = -k\int n(\mathbf{r})\ln n(\mathbf{r})\,d\mathbf{r}; cf. formula (3.3.13). Using these expressions, derive the equilibrium equations satisfied by the number density n(r)n(\mathbf{r}) and the total potential ϕ(r)\phi(\mathbf{r}), the latter being ϕext(r)+e∫n(r′)∣r−r′∣ dr′.\phi_{\text{ext}}(\mathbf{r}) + e\int\frac{n(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|}\,d\mathbf{r}'.

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