Articles of statistical mechanics

Evaluating Gibbs state in the second quantized formalism

First, let us fix our notation. If $A:\mathcal H \to \mathcal H$ is a linear operator on a single-particle Hilbert space $\mathcal H$, we can lift $A$ on the Fock space $\mathfrak F_{s/a}(\mathcal H)$ in the following two ways. Here the subscript indices $s/a$ stand for symmetrized/antisymmetrized versions corresponding to bosons/fermions respectively. Define $\Gamma, \text{d}\Gamma: […]

Connection between Boltzmann entropy and Kolmogorov entropy

What is the connectivity between Boltzmann's entropy expression and Shannon's entropy expression? mentions a realtionship between Shannon entropy and Bolltzmann entropy. Is there a relationship between Kolmogorov Sinai entropy and Boltzmann entropy? And kolmogorov entropy and Shannon entropy?

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor expression in the infinite domain of the vector $k$. I have tried to let $u$ in the direction of $k_z$ and then transform the current integral into a spherical coordinate with the following relation: $$ k_x = k […]

What is the connectivity between Boltzmann's entropy expression and Shannon's entropy expression?

What is the connection between Boltzmann’s entropy expression and Shannon’s entropy expression? Shannon’s entropy expression: $$ S= -K\sum_{i=1}^np_i\log (p_i) $$