Statistical Mechanics (multiplicity/accesbile microstates question)

[jeffreydk]

I am trying to show that the change in number of accessible microstates, and therefore the change in the multiplicity function $g$ of a simple system is

$$g=\left( \frac{\tau_F^2}{\tau_1\tau_2}\right)^{\frac{mC_V}{k_B}}$$

where the system is two identical copper blocks at fundamental temperatures $\tau_1, \tau_2$ brought into equilibrium through thermal contact, both with mass $m$ and heat capacity $C_V$. I've calculated the final temperature through

$$\Delta U = C_V(\tau_F-\tau_1)=C_V(\tau_2-\tau_F) \quad \Rightarrow \quad \tau_F=\frac{1}{2}(\tau_1+\tau_2)$$

I tried then to use that (with $\sigma=\log g$ as entropy)

$$\Delta \sigma = \left(\frac{\partial \sigma_1}{\partial U_1}\right)(\Delta U_1) + \left( \frac{\partial \sigma_2}{\partial U_2}\right)(-\Delta U_2)$$

But with this I get a result of

$$\Delta \sigma = \log g = 2C_V\left(\frac{\tau_F^2}{\tau_1\tau_2}-1\right)$$

But raising this with an exponential to find the change in $g$ would obviously not give me the desired result. Any help on where I'm going wrong with this would be greatly appreciated.

 

[LightHero]

Your mistake is in the use of the entropy formula. The formula you used is for a closed system, while this is an open system. For an open system, we need to consider the exchange of energy and entropy between two subsystems. In this case, we have two blocks of copper at different temperatures that are brought into thermal contact. The change in entropy is given by the following equation: \Delta \sigma = \sigma_1 + \sigma_2 - \sigma_{12} where \sigma_{12} is the combined entropy of the two blocks in thermal contact. This combined entropy can be calculated as \sigma_{12} = \sigma_1 + \sigma_2 + \frac{mC_V}{k_B}\log \left(\frac{\tau_F^2}{\tau_1\tau_2}\right) where \tau_F is the final temperature of the two blocks in equilibrium. Therefore, the change in g can be calculated as g = e^{\Delta \sigma} = e^{\sigma_1 + \sigma_2 - \sigma_{12}} = \left(\frac{\tau_F^2}{\tau_1\tau_2}\right)^{\frac{mC_V}{k_B}} as desired.

 

[astrokreng]

I understand your frustration and confusion in trying to understand this concept. Let's break down the problem and see where the issue lies.

First, let's define some terms to make sure we are on the same page. The multiplicity function, g, is a measure of the number of microstates that correspond to a given macrostate. In other words, it tells us how many different ways a system can be arranged to achieve a specific set of macroscopic properties, such as energy, volume, and number of particles.

In this problem, we have two identical copper blocks brought into thermal contact, with a change in energy, \Delta U, and a change in temperature, \Delta T. The final temperature, \tau_F, is given by the expression \tau_F = \frac{1}{2}(\tau_1 + \tau_2), as you correctly calculated.

Now, let's look at the expression you have for the change in multiplicity, \Delta g. You are using the equation \Delta \sigma = \left(\frac{\partial \sigma_1}{\partial U_1}\right)(\Delta U_1) + \left(\frac{\partial \sigma_2}{\partial U_2}\right)(-\Delta U_2), which is derived from the definition of entropy as \sigma = k_B \log g. However, this equation is only valid for reversible processes, where the system is in equilibrium at all times. In this case, the system is not in equilibrium during the entire process, as the two blocks are being brought into thermal contact and the temperature is changing. Therefore, we cannot use this equation to calculate the change in multiplicity.

Instead, we can use the definition of entropy as \Delta S = \frac{\Delta Q}{T}. Since the process is reversible, we can write this as \Delta S = \frac{\Delta U}{T}, where \Delta U is the change in internal energy and T is the final temperature. Using the expression for \tau_F that we calculated earlier, we can rewrite this as \Delta S = \frac{2C_V(\tau_2 - \tau_F)}{\tau_F}. Then, using the definition of entropy as \sigma = k_B \log g, we can write this as \Delta \sigma = \frac{2C_V}{k_B}\left(\frac{\tau_2}{\tau_F