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Number of States and Probabilities
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In order to systematize the study of a system using the method of state counting, we aim to establish a direct relationship between the probability of finding the system at a particular energy and the number of associated states.
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System in contact with a heat reservoir
Equation
Suppose a system with energy $E_r$ is in contact with a thermal reservoir of energy $E'$.
A thermal reservoir is understood as a system whose temperature remains constant. One way to achieve this is by having a large reservoir (like a water bath).
If both systems are isolated from the surroundings, the sum of their energies will remain constant, which can be expressed with as
| $E_0=E_r+E_h$ |
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This version provides clarity while retaining the essential information.
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Probability of finding the system in a state $r$
Equation
The probability of finding the system in a state where it has an energy $E_r$, while the reservoir has an energy $E' = E_0 - E_r$, is defined as
$P_r = C \cdot \Omega_r(E_r) \cdot \Omega'(E')$
where $C$ is a constant that is adjusted to ensure the probability is normalized.
Since $P_r$ represents the probability of finding the system in a particular state $r$, the number of states in state $r$ is equal to one. In other words, this implies that
$\Omega_r(E_r) = 1$
Therefore, the probability can be expressed with respect to as
| $P_r=C_h\Omega_h(E_0-E_r)$ |
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Normalization Condition
Equation
If we sum the probabilities of each state $r$, the result should be one. This signifies that it is normalized with the :
| $\sum_rP_r=1$ |
This is equivalent to stating that the system must necessarily be in one of the possible states.
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Development of the number of states in Taylor series
Equation
As the energy $E_r$ is much smaller than the total energy $E_0$, the logarithm of the number of states can be expanded around the energy $E_r$ as follows:
$\ln\Omega'(E_0-E_r)=\ln\Omega'(E_0)-\left.\displaystyle\frac{\partial\Omega'}{\partial E'}\right\vert_0E_r\ldots$
Since the derivative of the logarithm of the number of states is equal to the beta function:
$\beta=\left.\displaystyle\frac{\partial\Omega'}{\partial E'}\right\vert_0$
It follows that, in a first approximation with ,
| $\ln\Omega_h(E_0-E_r)=\ln\Omega_h(E_0)-\beta E_r$ |
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Equation for the Probability of State $r$
Equation
If we substitute the expression
| $\ln\Omega_h(E_0-E_r)=\ln\Omega_h(E_0)-\beta E_r$ |
with beta del sistema $1/J$, energía del estado $r$ $J$, energía del sistema $J$ and número de Estados $-$
into the equation for probability with constante de Normalización $-$, energía del estado $r$ $J$, energía del sistema $J$, número de Estados $-$ and probabilidad del estado $r$ $-$,
| $P_r=C_h\Omega_h(E_0-E_r)$ |
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we obtain with constante de Normalización $-$, energía del estado $r$ $J$, energía del sistema $J$, número de Estados $-$ and probabilidad del estado $r$ $-$ the probability
| $P_r=Ce^{-\beta E_r}$ |
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where $C$ is a constant to be determined using the normalization condition.
The expression $e^{-\beta E}$ is referred to as the Boltzmann factor, and the distribution it describes is known as the canonical distribution.
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Normalization constant
Equation
Under the normalization condition with estado $r$ $-$ and probabilidad del estado $r$ $-$,
| $\sum_rP_r=1$ |
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it is obtained that the normalization constant $C$ is equal to estado $r$ $-$ and probabilidad del estado $r$ $-$:
| $C^{-1}=\sum_re^{-\beta E_r}$ |
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