玻尔兹曼分布

The Boltzmann distribution is a cornerstone of statistical mechanics and is arguably its most useful result for practical applications. It can be derived by considering a small system in thermal contact with a large heat reservoir. The combined system (system + reservoir) is isolated, and by applying Boltzmann’s entropy principle ([latex]S = k_B \ln W[/latex]) to the reservoir, one can find the most probable energy distribution for the small system. The result is that the probability of the system being in state ‘i’ with energy [latex]E_i[/latex] is [latex]P_i \propto e^{-E_i/k_B T}[/latex].

The term [latex]k_B T[/latex] represents the characteristic thermal energy available at temperature T. The ratio [latex]E/k_B T[/latex] is dimensionless and determines the probability. If a state’s energy E is much less than the thermal energy ([latex]E \ll k_B T[/latex]), the exponential factor is close to 1, and the state is highly probable. If the energy is much greater than the thermal energy ([latex]E \gg k_B T[/latex]), the factor is very small, and the state is very unlikely to be occupied. This exponential dependence is responsible for many phenomena, such as the rapid increase in chemical reaction rates with temperature, as more molecules possess the necessary activation energy.