Physlib.StatisticalMechanics.MicroCanonicalEnsemble.ThermoQuantities

For a microcanonical Hamiltonian $H$ with extrinsic parameters $d \in D$, the partition function $Z(\beta)$ is a real-valued function that maps a thermodynamic beta $\beta \in \mathbb{R}$ to the sum over the finite state space $H.\text{dim}(d)$ of the Boltzmann factors $e^{-\beta E}$.

For a microcanonical Hamiltonian $H$ and a parameter $d \in D$, let $H.\text{dim}(d)$ be the finite set of microstates. The complexified partition function $Z(z)$ is a function that maps a complex number $z \in \mathbb{C}$ to a complex value, defined as the Laplace transform of the energy distribution over the microstates. In the discrete case, it corresponds to the sum $Z(z) = \sum_{\omega \in H.\text{dim}(d)} e^{-z E(\omega)}$, where $E(\omega)$ is the energy associated with the state $\omega$.

For a microcanonical Hamiltonian $H$ with parameter $d \in D$, this is the set of complex numbers $z \in \mathbb{C}$ for which the Laplace transform defining the complex partition function $Z(z)$ converges.

For a microcanonical Hamiltonian $H$ with parameters $d$, let $Z(\beta)$ denote the partition function and let $\mathcal{D}_{\mathbb{C}}$ be the domain of convergence for its complex-valued extension. If a real thermodynamic beta $\beta$ belongs to the interior of $\mathcal{D}_{\mathbb{C}}$ when embedded in the complex plane, then the partition function $Z$ is infinitely differentiable ($C^\infty$) at $\beta$.

For a microcanonical Hamiltonian $H$ with extrinsic parameters $d$, the function $Z(T)$ represents the partition function as a function of the temperature $T \in \mathbb{R}$. This function is defined by substituting the thermodynamic beta $\beta$ with $1/T$ (assuming units where $k_B = 1$) in the standard partition function $Z(\beta) = \sum e^{-\beta E}$.

For a given Hamiltonian $H$ with extrinsic parameters $d$, the internal energy $U(\beta)$ is defined as the negative derivative of the natural logarithm of the partition function $Z(\beta)$ with respect to the thermodynamic beta $\beta$: $$U(\beta) = -\frac{\partial \ln Z(\beta)}{\partial \beta}$$ where $\beta$ represents the inverse temperature $1/k_B T$.

For a microcanonical Hamiltonian, the Helmholtz free energy $A(T)$ (also denoted as $F$) is defined as a function of the temperature $T \in \mathbb{R}$ by the relation: $$A(T) = -T \ln Z(T)$$ where $Z(T)$ is the partition function evaluated at temperature $T$.

For a microcanonical system, the entropy $S(T)$ is defined as a function of the temperature $T \in \mathbb{R}$ by the negative derivative of the Helmholtz free energy $A(T)$ with respect to temperature: $$S(T) = -\frac{\partial A(T)}{\partial T}$$ where $A(T)$ is the Helmholtz free energy.

For a microcanonical Hamiltonian, the entropy $S(\beta)$ is defined as a function of the thermodynamic beta $\beta$ by the relation: $$S(\beta) = \ln Z(\beta) + \beta U(\beta)$$ where $Z(\beta)$ is the partition function and $U(\beta)$ is the internal energy.

For a system described by the $NVE\text{Hamiltonian}$ (where $N$ is the number of particles and $V$ is the volume), the pressure $P$ at temperature $T \in \mathbb{R}$ is defined as the thermodynamic conjugate variable to the volume $V$. It is given by the negative partial derivative of the Helmholtz free energy $A$ with respect to the volume: $$P(T) = -\left( \frac{\partial A}{\partial V} \right)_{T, N}$$ where $A(T)$ is the Helmholtz free energy defined by $A(T) = -T \ln Z(T)$.