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To model systems using statistical mechanics, we need to investigate how statistical ensembles can be influenced by the parameters that describe the macroscopic system. For particles, temperature is established as a parameter that reflects whether systems are in equilibrium, maintaining their energies at a constant level.

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We can study what happens when we put two systems of particles in contact in such a way that they can exchange energy but not particles.

Let's also assume that the system is isolated from the surroundings, meaning it has a total energy of $E_0$.

Suppose initially the first system has an energy of $E$, which is associated with $\Omega(E)$ states.

Since the total energy is $E_0$, the second system can only have the energy $E_0-E" and a number of associated states $\Omega(E_0-E)$.

Once we bring them into contact, they can exchange energy until they reach some equilibrium. In this regard, the value of $E$ will vary, and the probability of finding the systems such that the first one has a value of $E$ will also vary.

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Each system $\Omega$ has a number of possible states that depend on its energy $E$. Therefore, if the system we are studying has an energy $E$, the number of possible states will be $\Omega(E)$.

The system under study is in contact with a reservoir that provides energy $E$, so the total energy is $E_0$ minus the immersed system's energy, $E$. Therefore, the reservoir has $\Omega(E_0 - E)$ possible states. The probability of finding the total system with an energy $E$ in the immersed system is expressed as the product of the number of states with :

where $C$ is a normalization constant. The energy $E$ will be the one for which the probability is maximum.

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When we compare how the number of states varies with energy $E$, we observe that the behavior of the system and the reservoir is opposite:

This happens because as the energy increases, the energy of the reservoir decreases, leading to a reduction in the number of states it can access.

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When we multiply the number of cases, we obtain a function with a very pronounced peak.

The system is more likely to be found at the energy where the peak of the probability curve occurs.

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If the probability of two isolated systems, each with a total energy of $E_0$ and one of the systems having an energy of $E$, is given by factor de normalización $-$, numero de estados del reservorio con energía $E_0-E$ $-$, numero de estados del sistema con la energía $E$ $-$ and probabilidad del sistema de tener una energía $E$ $-$

We can estimate the probable energy $E$ at which they will be found by seeking the maximum probability. To do this, we need to take the derivative with respect to energy $E$ and set the derivative equal to zero.

$\displaystyle\frac{\partial P}{\partial E} = \displaystyle\frac{\partial\Omega}{\partial E}\Omega' + \Omega\displaystyle\frac{\partial\Omega'}{\partial E} = 0$

If we divide the expression by $\Omega\Omega'$ and replace the energy difference $E_0-E$ with $E'$, we can rewrite the condition to determine the most probable situation as follows:

If there is a probability $P(E)$ of finding

$\displaystyle\frac{1}{\Omega}\displaystyle\frac{\partial\Omega}{\partial E} - \displaystyle\frac{1}{\Omega'}\displaystyle\frac{\partial\Omega'}{\partial E'} = 0$

The negative sign arises from the change of variables, as with

$E' = E_0-E$

the derivative with respect to $E'$ results in factor de normalización $-$, numero de estados del reservorio con energía $E_0-E$ $-$, numero de estados del sistema con la energía $E$ $-$ and probabilidad del sistema de tener una energía $E$ $-$

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When a system is in contact with an energy reservoir $E_0$, it is likely to be found with an energy $E$ for which the probability with factor de normalización $-$, numero de estados del reservorio con energía $E_0-E$ $-$, numero de estados del sistema con la energía $E$ $-$ and probabilidad del sistema de tener una energía $E$ $-$

reaches its maximum. The energy can be determined by taking the derivative of this expression with respect to energy $E$ and setting it equal to zero. This is equivalent to taking the derivative of the logarithm of the probability:

$\ln P(E) = \ln C + \ln\Omega(E) + \ln\Omega(E_0-E)$

Leading to:

$\displaystyle\frac{\partial\ln\Omega}{\partial E} + \displaystyle\frac{\partial\ln\Omega}{\partial E} = 0$

If we make a change of variable:

$E' = E_0 - E$

We obtain the equilibrium condition with factor de normalización $-$, numero de estados del reservorio con energía $E_0-E$ $-$, numero de estados del sistema con la energía $E$ $-$ and probabilidad del sistema de tener una energía $E$ $-$:

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The equilibrium condition of a system in contact with a reservoir is expressed with energía del reservorio $J$, energía del sistema $J$, numero de estados del reservorio con energía $E'$ $-$ and numero de estados del sistema con la energía $E$ $-$

This allows us to introduce a function $\beta$ with energía del reservorio $J$, energía del sistema $J$, numero de estados del reservorio con energía $E'$ $-$ and numero de estados del sistema con la energía $E$ $-$ in the following way:

This function characterizes the state of the system and becomes relevant when the system is in equilibrium with another system.

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When a system is in contact with an energy reservoir $E_0$, it is likely to be found with an energy $E$ for which, with factor de normalización $-$, numero de estados del reservorio con energía $E_0-E$ $-$, numero de estados del sistema con la energía $E$ $-$ and probabilidad del sistema de tener una energía $E$ $-$, the probability

reaches its maximum. The energy can be determined by taking the derivative of this expression with respect to energy $E$ and setting it equal to zero. This is equivalent to taking the derivative of the logarithm of the probability:

$\ln P(E) = \ln C + \ln\Omega(E) + \ln\Omega(E_0-E)$

Thus, with energía del reservorio $J$, energía del sistema $J$, numero de estados del reservorio con energía $E'$ $-$ and numero de estados del sistema con la energía $E$ $-$, we have

If we perform a change of variable

$E' = E_0 - E$

we obtain the equilibrium condition with energía del reservorio $J$, energía del sistema $J$, numero de estados del reservorio con energía $E'$ $-$ and numero de estados del sistema con la energía $E$ $-$:

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If we assume that we find the system at the energy for which the probability is maximum, we can associate this with the equilibrium situation of a system, where the probability is maximum.

On the other hand, we know that two systems are in thermal equilibrium if their temperatures are equal. Therefore, the fact that the functions $\beta$ are equal suggests that $\beta$ is related to temperature.

Since the units of $\beta$ are the reciprocal of energy, we can define it as follows with :

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By introducing the relationship with beta del reservorio $1/J$, constante de Boltzmann $J/K$ and temperatura del sistema $K$

the equilibrium condition with beta del reservorio $1/J$ and beta del sistema $1/J$

is simplified to just

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Video: Balance Condition and Temperature