"In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution)..."
A probability distribution used to determine the state of a system in thermal equilibrium.
Thermodynamics: This is the study of heat transfer and its relationship to other forms of energy. Thermodynamics lays the foundation for statistical mechanics, which is the study of the behavior of atoms and molecules in a system.
Phase transitions: Phase transitions occur when a material undergoes a change in its physical state, such as from solid to liquid, or liquid to gas. The behavior of a material near a phase transition is a key aspect of statistical mechanics.
Probability theory: Probability theory provides the mathematical means to describe random events, and is used to analyze the behavior of systems of particles in statistical mechanics.
Statistical ensembles: Statistical ensembles are collections of systems or particles that are characterized by their macroscopic properties. The most commonly used ensembles are the microcanonical, canonical, and grand canonical ensembles.
Partition function: The partition function is a mathematical function that describes the energy levels of a system of particles, and is used to calculate the probability that the system is in a given state.
Gibbs entropy: Gibbs entropy is a measure of the amount of disorder in a system, and is a key concept in statistical mechanics.
Boltzmann distribution: The Boltzmann distribution is a probability distribution that describes the distribution of particles in a system at a given temperature.
Maxwell–Boltzmann distribution: The Maxwell–Boltzmann distribution describes the distribution of velocities of particles in a gas at a given temperature.
Bose–Einstein distribution: The Bose–Einstein distribution describes the distribution of particles that obey Bose–Einstein statistics, which applies to particles with integer spin, such as photons.
Fermi–Dirac distribution: The Fermi–Dirac distribution describes the distribution of particles that obey Fermi–Dirac statistics, which applies to particles with half-integer spin, such as electrons.
Quantum mechanics: Quantum mechanics is the study of the behavior of particles at the nanoscale, and is necessary to describe the behavior of particles in statistical mechanics.
Statistical mechanics applications: Statistical mechanics has applications in a wide range of fields, including materials science, chemistry, and astrophysics.
Canonical Ensemble: This distribution describes the probability of a particle having a certain amount of energy when the number of particles and the volume of the system are fixed. In this situation, the system is said to be in thermal equilibrium with a heat bath at a given temperature, and the distribution is known as the canonical ensemble.
Grand Canonical Ensemble: This distribution describes the probability of a particle having a certain amount of energy when neither the number of particles nor the volume of the system is fixed. In this situation, the system is said to be in a state of thermodynamic equilibrium with a heat bath and a particle reservoir at a given temperature, and the distribution is known as the grand canonical ensemble.
"...a probability distribution or probability measure that gives the probability that a system will be in a certain state..."
"The distribution is expressed in the form: p_i ∝ exp(-ε_i/kT), where pi is the probability of the system being in state i, εi is the energy of that state, and a constant kT of the distribution is the product of the Boltzmann constant k and thermodynamic temperature T."
"Therefore the Boltzmann distribution can be used to solve a wide variety of problems."
"The distribution shows that states with lower energy will always have a higher probability of being occupied."
"The ratio of probabilities of two states is known as the Boltzmann factor..."
"The Boltzmann factor characteristically only depends on the states' energy difference."
"The Boltzmann distribution is named after Ludwig Boltzmann..."
"Ludwig Boltzmann first formulated it in 1868..."
"...during his studies of the statistical mechanics of gases in thermal equilibrium."
"The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902."
"The Boltzmann distribution should not be confused with the Maxwell–Boltzmann distribution or Maxwell-Boltzmann statistics."
"The Maxwell-Boltzmann distributions give the probabilities of particle speeds or energies in ideal gases."
"The distribution of energies in a one-dimensional gas, however, does follow the Boltzmann distribution."
"The symbol ∝ denotes proportionality (see § The distribution for the proportionality constant)."
"...a Boltzmann distribution (also called Gibbs distribution) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system."
"...a constant kT of the distribution is the product of the Boltzmann constant k and thermodynamic temperature T."
"States with lower energy will always have a higher probability of being occupied."
"The Boltzmann factor...depends on the states' energy difference."
"Therefore the Boltzmann distribution can be used to solve a wide variety of problems."