Information about Thermodynamic Beta
In statistical mechanics, the thermodynamic beta is a numerical quantity related to the thermodynamic temperature of a system. The thermodynamic beta can be viewed as a connection between the statistical interpretation of a physical system and thermodynamics.
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:
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.
Details
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 microstates 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 temperature 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
Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force.
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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
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Thermodynamics (from the Greek θερμη, therme, meaning "heat" and δυναμις, dynamis, meaning "power") is a branch of physics that studies the effects of changes in temperature, pressure, and volume on
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microstate (or ministate) is a sovereign state having a very small population or very little land area – usually both. Perhaps the most well-known examples of microstates include Andorra, Liechtenstein, Malta, Monaco, San Marino, Nauru, Palau, Tuvalu, Saint Kitts and
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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
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The Boltzmann constant (k or kB) is the physical constant relating temperature to energy.
It is named after the Austrian physicist Ludwig Boltzmann, who made important contributions to the theory of statistical mechanics, in which this
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It is named after the Austrian physicist Ludwig Boltzmann, who made important contributions to the theory of statistical mechanics, in which this
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Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each respectively possess energy Ei:
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A canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system. The probability distribution is characterised by the proportion pi
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