Canonical/grand canonical ensemble

In summary, the conversation discusses a system described by a one-dimensional asymmetric double well potential. The potential energies for each well can be approximated by harmonic potentials. The average number of particles in each well at a given temperature is calculated using the canonical partition function and the relation between the canonical and grand canonical partition functions. The conditions for having equal populations in each well at a certain temperature are derived, but there are concerns about the integrals used in the calculation. Further exploration is needed to solve the problem.
  • #1
Brian-san
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Homework Statement


Some systems are adequately described by a one-dimensional potential in the form of an asymmetric double well. To good accuracy each can assumed to be harmonic with potential energies:
[tex]V_L(x)=\frac{1}{2}k_Lx^2, V_R(x)=\epsilon+\frac{1}{2}k_R(x-a)^2[/tex]
Here, [itex]\epsilon=V_R(a)>0[/itex]. [itex]N[/itex] classical particles of mass [itex]m[/itex] are brought into thermal equilibrium in this potential.

a) At temperature [itex]T[/itex] what is the average number of particles in each well?

b) What conditions need to be imposed on the parameters of the potential in order to have equal populations at temperature [itex]T^*[/itex]?

c) Calculate the difference, [itex]u_R-u_L[/itex], between the internal energy of two particles in the two wells. Explain why your result is not inconsistent with the way the particles are distributed between the wells for [itex]T\geq T^*[/itex] (when the conditions found in (b) hold.)

Homework Equations


Relation between canonical and grand canonical partition functions:
[tex]\Xi=\sum_{N=0}^{\infty}z^NQ_N[/tex]

Canonical partition function for 1 particle/oscillator:
[tex]Q_1=\int e^{-\beta H(p,q)}dpdq[/tex]

Average number of particles:
[tex]\langle N\rangle=z\frac{\partial}{\partial z}ln\Xi[/tex]

The Attempt at a Solution


Assuming the N oscillators are non interacting, [itex]Q_N=Q_1^N[/itex], so then
[tex]\Xi=\sum_{N=0}^{\infty}z^NQ_N=\sum_{N=0}^{\infty}(zQ_1)^N=\frac{1}{1-zQ_1}[/tex]

Then the average number of particles is simply
[tex]\langle N\rangle=z\frac{\partial}{\partial z}ln\Xi=-z\frac{\partial}{\partial z}ln(1-zQ_1)=\frac{zQ_1}{1-zQ_1}=\frac{1}{(zQ_1)^{-1}-1}[/tex]

For the left well,
[tex]Q_1=\frac{1}{h}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\beta(\frac{p^2}{2m}+\frac{k_Lx^2}{2})}dxdp=\frac{1}{h}\int_{-\infty}^{\infty}e^{-\frac{\beta p^2}{2m}}dp\int_{-\infty}^{\infty}e^{-\frac{\beta k_Lx^2}{2}}dx=\frac{1}{h}\sqrt{\frac{2\pi m}{\beta}}\sqrt{\frac{2\pi}{\beta k_L}}=\frac{2\pi}{h\beta}\sqrt{\frac{m}{k_L}}=\frac{k_BT}{\hbar\omega_L}[/tex]

using [itex]\beta=\frac{1}{k_BT}, k_L=m\omega_L^2[/itex]. Then I got
[tex]\langle N_L\rangle=\frac{1}{\frac{\hbar\omega_L}{k_BT}e^{-\frac{\mu}{k_BT}}-1}[/tex]

By a similar process for the right well I found
[tex]\langle N_R\rangle=\frac{1}{\frac{\hbar\omega_R}{k_BT}e^{\frac{\epsilon-\mu}{k_BT}}-1}[/tex]

This led to the condition for part b that when [itex]T=T^*, \langle N_L\rangle=\langle N_R\rangle[/itex] the relation between the parameters is
[tex]\omega_L=\omega_Re^{\frac{\epsilon}{k_BT^*}}, k_L=k_Re^{\frac{2\epsilon}{k_BT^*}}[/tex]

However, I went back and asked the professor about this as my functions for average number of particles could produce negative values for certain temperature ranges (usually as T approached infinity). I was then told that I couldn't perform the dx integral of my partition function over all x. Obviously there is an intersection point between the two parabolas between 0 and a (say it occurs at x=c, 0<c<a), thus the left well integral covers the interval [itex](-\infty,c][/itex] and the right covers [itex][c,\infty)[/itex].

Trying to do the integrals in this manner proved more difficult as the integrands were Gaussian so there wouldn't be an exact closed form over the new intervals (I did find it could be expressed in terms of error functions, but that seems to make things more complicated). I was also told that this could be solved solely in the canonical ensemble, but I don't see exactly how. The particle number in each well isn't fixed as they are allowed to move between the different wells.
 
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  • #2
The only way I could think of is to calculate the partition function for each configuration (n, N-n) and then sum them up. But this would still require finding the integrals in a new manner, which I am not sure how to do. Any help is appreciated.
 

1. What is the difference between canonical and grand canonical ensemble?

The canonical ensemble describes a system with a fixed number of particles, volume, and temperature. The grand canonical ensemble describes a system where the number of particles is allowed to fluctuate, but the volume and temperature remain fixed.

2. What is the purpose of using a canonical/grand canonical ensemble?

The canonical/grand canonical ensemble is used to describe the statistical behavior of a large number of particles in a system. It allows us to calculate the average properties of the system, such as energy and temperature, and understand how these properties change as the system evolves.

3. How are the canonical and grand canonical ensembles related?

The grand canonical ensemble can be thought of as a combination of many canonical ensembles, each with a different number of particles. This allows us to describe systems with varying number of particles, while still maintaining a fixed temperature and volume.

4. What are the main assumptions of the canonical/grand canonical ensemble?

The main assumptions of the canonical/grand canonical ensembles are that the particles in the system are in thermal equilibrium, and that the interactions between particles are weak enough to be treated as independent.

5. How is the partition function calculated in the canonical/grand canonical ensemble?

In the canonical ensemble, the partition function is calculated as the sum over all possible states of the system. In the grand canonical ensemble, it is calculated as the sum over all possible states and number of particles. Both ensembles use the Boltzmann distribution to weight the contributions of each state.

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