jagamisfunktsiooni

Jagamisfunktsiooni, known in English as the partition function, is a fundamental concept in statistical mechanics and thermodynamics. It is a mathematical object that encodes all the thermodynamic information about a system in thermal equilibrium. Specifically, for a system with discrete energy levels $E_i$ and corresponding degeneracies $g_i$, the canonical partition function $Z$ is defined as the sum over all accessible states, weighted by the Boltzmann factor: $Z = \sum_i g_i e^{-E_i/k_B T}$, where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.

The partition function serves as a bridge between the microscopic properties of a system (its energy levels) and its macroscopic thermodynamic properties. For instance, the Helmholtz free energy $F$ of the system can be directly obtained from the partition function through the relation $F = -k_B T \ln Z$. From the Helmholtz free energy, other thermodynamic quantities such as internal energy ($U$), entropy ($S$), and pressure ($P$) can be derived. For example, the internal energy is given by $U = - (\frac{\partial \ln Z}{\partial (1/T)})_V$ and the entropy by $S = - (\frac{\partial F}{\partial T})_V$.

The partition function is particularly useful in systems where the number of possible states is very large, making direct calculation of macroscopic properties computationally challenging. By providing a compact representation of the system's state space and its thermal behavior, the partition function simplifies thermodynamic calculations and allows for predictions of how a system will respond to changes in temperature, volume, or other external conditions. It is a cornerstone for understanding phase transitions, chemical equilibria, and the behavior of gasses, liquids, and solids at a microscopic level.