Luiza Angheluta
<aside> 📖 Boltzmann statistics refers to an ensemble of microstates for a statistical system that exchanges heat with its environment at a fixed temperature $T$. Such a collection of ensemble of microstates is also called the canonical ensemble. At equilibrium, the system has the same temperature as its thermal bath. Thus, we control the temperature of the system by changing the temperature of its thermal bath. This is much more attainable experimentally than tuning directly the system’s internal energy $U$ or entropy $S$. In the equilibrium state, the thermodynamic quantities ($T,S,U,P,V,\cdots$) are related to each other through equations of state independent on how the system approaches this equilibrium state. However, it is very useful for us to choose the path to equilibrium that is most suitable for experimental or theoretical investigations of the system. In other words, the protocol of keeping T fixed and letting U vary is different than the protocol of letting T vary while keeping U fix. The later protocol samples equally-likely microstates from an isolated systems where their internal energy is fixed (controlled). Here is a recapitulation of the two-state model for isolated systems. By contrast, the first protocol samples the canonical ensembles where a microstate’s probability depends on both the temperature and the energy associated to the microstate**.** For instance, a system at low temperature is more likely to be in low-energy states (closer to the ground state), while a system at higher temperature which is more likely to be in high-energy states. The probability that the system is in a given microstate is determined by the Boltzmann distribution which is given in terms of the Boltzmann factor $e^{- E/kT}$, where $E$ is the total internal energy of a given microstate, which fluctuates from one microstate to the other.
In this lecture, we will:
Further readings on the 2021 Nobel Price in Physics related to statistical mechanics and the physics of complex systems.
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