Thermodynamic beta
Encyclopedia
In statistical mechanics
, the thermodynamic beta is a physical quantity related to the thermodynamic temperature
of a system. It can be calculated from formula
where
is the Boltzmann constant. The thermodynamic beta can be viewed as a connection between the statistical interpretation of a physical system and thermodynamics
. It is sometimes considered a more fundamental quantity than temperature.
of each system will be denoted by Ω1 and Ω2. Under our assumptions Ωi depends only on Ei. Thus the number of microstates for the combined system is
We will derive β from the following fundamental assumption:
(In other words, the system naturally seeks the maximum number of microstates.) Therefore, at equilibrium,
But E1 + E2 = E implies
So
i.e.
The above relation motivates the definition of β:
T. Thus intuitively one would expect that β be related to T in some way. This link is provided by the formula
where k is the Boltzmann constant. So
Substituting into the definition of β gives
Comparing with the thermodynamic formula
we have
where
is sometimes called the fundamental temperature of the system with units of energy.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, the thermodynamic beta is a physical quantity related to the thermodynamic temperature
Thermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
of a system. It can be calculated from formulawhere
is the Boltzmann constant. The thermodynamic beta can be viewed as a connection between the statistical interpretation of a physical system and thermodynamicsThermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
. It is sometimes considered a more fundamental quantity than temperature.
Statistical interpretation
From the statistical point of view, β is a numerical quantity relating two macroscopic systems in equilibrium. The exact formulation is as follows. Consider two systems, 1 and 2, in thermal contact, with respective energies E1 and E2. We assume E1 + E2 = some constant E. The number of microstatesMicrostate (statistical mechanics)
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...
of each system will be denoted by Ω1 and Ω2. Under our assumptions Ωi depends only on Ei. Thus the number of microstates for the combined system is
We will derive β from the following fundamental assumption:
- When the combined system reaches equilibrium, the number Ω is maximized.
(In other words, the system naturally seeks the maximum number of microstates.) Therefore, at equilibrium,
But E1 + E2 = E implies
So
i.e.
The above relation motivates the definition of β:
Connection with thermodynamic view
On the other hand, when two systems are in equilibrium, they have the same thermodynamic temperatureThermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
T. Thus intuitively one would expect that β be related to T in some way. This link is provided by the formula
where k is the Boltzmann constant. So
Substituting into the definition of β gives
Comparing with the thermodynamic formula
we have
where
is sometimes called the fundamental temperature of the system with units of energy.See also
- Boltzmann factorBoltzmann factorIn physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...
- Boltzmann distributionBoltzmann distributionIn chemistry, physics, and mathematics, the Boltzmann distribution is a certain distribution function or probability measure for the distribution of the states of a system. It underpins the concept of the canonical ensemble, providing its underlying distribution...
- Canonical ensembleCanonical ensembleThe canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...
- Ising modelIsing modelThe Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...