Boltzmann factor
Encyclopedia
In physics
, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state
in a multi-state system in thermodynamic equilibrium
at temperature
. The Boltzmann factor is normally used when discussing systems described by the canonical ensemble
- for the grand canonical ensemble
, it is more appropriate to use the Gibbs' factor which takes into account particle transfer between the system and the environment.
The probability that a system is in a state
is given by:
Where
is given by
is the Partition Function
,
is Boltzmann's Constant, 
is temperature
and
is the energy
of state
.
The Boltzmann factor is the term:
. The system is in contact with a heat reservoir and thus conservation of energy gives total energy
with
being total system energy and 
being total reservoir energy. In equilibrium, the number of states in 
and 
is the multiplicity 
. Similar to total energy, we can write:
The probability that an atom is in state
is related to the number of states in the reservoir - from Equipartition theorem
. Consider taking the ratio of two probabilities:
We can relate the number of states to the entropy
via
giving:
The Fundamental thermodynamic relation tells us that for the reservoir (rearranging and neglecting the chemical potential term):
Where
is the entropy, 
is the internal energy, 
is pressure and 
is volume.
For a gas, it is reasonable to assume that
so this term also drops out giving:
By energy conservation
and 
giving
Substituting into the probability ratio gives:
Where we have defined an arbitrary symbol
, which is the reciprocal of Boltzmann's constant times temperature. Through separation of variables, we can write
and hence
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state
in a multi-state system in thermodynamic equilibriumThermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
at temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
. The Boltzmann factor is normally used when discussing systems described by the canonical ensembleCanonical ensemble
The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...
- for the grand canonical ensemble
Grand canonical ensemble
In statistical mechanics, a grand canonical ensemble is a theoretical collection of model systems put together to mirror the calculated probability distribution of microscopic states of a given physical system which is being maintained in a given macroscopic state...
, it is more appropriate to use the Gibbs' factor which takes into account particle transfer between the system and the environment.
The probability that a system is in a state
is given by:Where
is given by is the Partition FunctionPartition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
,
is Boltzmann's Constant, is temperatureTemperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
and
is the energyEnergy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of state
.The Boltzmann factor is the term:
Derivation
Consider a single atom system with energy states. The system is in contact with a heat reservoir and thus conservation of energy gives total energywith
being total system energy and being total reservoir energy. In equilibrium, the number of states in and is the multiplicity . Similar to total energy, we can write:The probability that an atom is in state
is related to the number of states in the reservoir - from Equipartition theoremEquipartition theorem
In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...
. Consider taking the ratio of two probabilities:
We can relate the number of states to the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
via
giving:
The Fundamental thermodynamic relation tells us that for the reservoir (rearranging and neglecting the chemical potential term):
Where
is the entropy, is the internal energy, is pressure and is volume.For a gas, it is reasonable to assume that
so this term also drops out giving:By energy conservation
and givingSubstituting into the probability ratio gives:
Where we have defined an arbitrary symbol
, which is the reciprocal of Boltzmann's constant times temperature. Through separation of variables, we can writeand hence