One-particle system with phase transition?

[hilbert2]

Hi, I'm Teemu from Finland, and I'm just getting my master's degree in physics completed (only
one course in electronics left). This is my first post in Physicsforums.

I was thinking about a problem in statistical mechanics. As you all know, if we have a one-
particle system where the particle is subject to some potential V(x), we can solve the particle's
energy spectrum from the Schrodinger equation. When we know the energy spectrum, we can form
the statistical mechanical partition function Z. Then we can calculate E(T), the expectation value
of the systems energy by differentiating log(Z) with respect to temperature. From this we finally
get the heat capacity C(T) by differentiating again.

If a system has a first-order phase transition at some particular temperature, the function E(T)
has a discontinuity at that temperature. Phase transitions are generally only thought to occur
at the thermodynamic limit (i.e. in a system of very many particles). But to me, it seems to be
only a matter of imagining a system with an appropriate energy spectrum to get such behavior to
occur in a one-particle system.

Denote the energy spectrum function with E(n), that specifies an energy for every integer n.
In the case of continuous energy spectrum, we would use function g(E), density of states, instead.

What kind of a function E(n) would lead to a partition function that would have discontinuous
first derivative (or 'almost' discontinuous, jumping very sharply but continuously at some value
of T)? Is it possible to imagine a potential energy function V(x) that would cause a single
particle bound by that potential to have that kind of an energy eigenvalue spectrum?

 

[king vitamin]

If your potential is temperature independent, I can't see how your free energy could be a nonanalytic function of temperature. At T = 0, you simply have the particle in its ground state (F = E_gnd), and as one increases temperature, one simply finds that the excited states have some finite probability of being occupied and F should be smooth.

 

[hilbert2]

If your potential is temperature independent, I can't see how your free energy could be a nonanalytic function of temperature.

Yes, it can't be exactly nonanalytic, but one can approach a nonanalytic behavior with a sequence of analytic functions (think about function f(x) = exp(-x²ⁿ) when integer n grows without bound).

The derivative of the partition function could have a very sharp but continuous jump at some value of T, approximating a 1st order phase transition.

 

[hilbert2]

I'll put the problem more exactly so its easier to grasp:

Usually we know the set of energies: {E₀, E₁, E₂, ...} that a system could have. Then we can form the partition function as the sum of Boltzmann factors:

Z(T) = \sum_nexp(-E_n/k_bT) .

This is what we usually do. But now I want to do the other way around, I first have some partition function
Z(T) and I want to deduce what set of energies generates it. I can't find any literature handling this kind of a problem.

If I could do this, I could construct a partition function that has a very rapid change in derivative around some value of T, and then deduce the energy spectrum.

 

[hilbert2]

Sorry for bumping this thread, but I think I found the correct inversion formula...

Suppose the partition function is

Z(T)=\int^{\infty}_{-\infty}dEg(E)exp\left(-\frac{E}{kT}\right)

for some density of states g(E). Now we can obtain function g(E) as following:

\frac{1}{2\pi}\int^{\infty}_{-\infty}dzexp(iEz)Z\left(\frac{1}{ikz}\right)=\frac{1}{2\pi}\int^{\infty}_{-\infty}dE'\int^{\infty}_{-\infty}dzg(E')exp(-iE'z)exp(iEz)=\int^{\infty}_{-\infty}dE'\delta(E-E')g(E')=g(E)

Here the formula \int^{\infty}_{-\infty}dzexp(-iE'z)exp(iEz)=2\pi\delta(E-E') was used.

I found an article from the 1930's that deals with this problem, lol... http://jcp.aip.org/resource/1/jcpsa6/v7/i12/p1097_s1?isAuthorized=no

 

[RonL]

Hi, I'm Teemu from Finland, and I'm just getting my master's degree in physics completed (only
one course in electronics left). This is my first post in Physicsforums.

I was thinking about a problem in statistical mechanics. As you all know, if we have a one-
particle system where the particle is subject to some potential V(x), we can solve the particle's
energy spectrum from the Schrodinger equation. When we know the energy spectrum, we can form
the statistical mechanical partition function Z. Then we can calculate E(T), the expectation value
of the systems energy by differentiating log(Z) with respect to temperature. From this we finally
get the heat capacity C(T) by differentiating again.

If a system has a first-order phase transition at some particular temperature, the function E(T)
has a discontinuity at that temperature. Phase transitions are generally only thought to occur
at the thermodynamic limit (i.e. in a system of very many particles). But to me, it seems to be
only a matter of imagining a system with an appropriate energy spectrum to get such behavior to
occur in a one-particle system.

Denote the energy spectrum function with E(n), that specifies an energy for every integer n.
In the case of continuous energy spectrum, we would use function g(E), density of states, instead.

What kind of a function E(n) would lead to a partition function that would have discontinuous
first derivative (or 'almost' discontinuous, jumping very sharply but continuously at some value
of T)? Is it possible to imagine a potential energy function V(x) that would cause a single
particle bound by that potential to have that kind of an energy eigenvalue spectrum?

Teemu, this is very much over my head, but I think there are some things in these pages that might help in regard to your questions.

http://arxiv.org/abs/1207.0434

 

[Avodyne]

1) hilbert2 is effectively using the inverse Laplace transform.

2) One-particle systems do not have phase transitions. That requires a large number of particles (strictly speaking, an infinite number).

 

[hilbert2]

^ Yes, the inversion of Z(T) can be done with the Bromwich integral.

I can form some function Z(T) that has a discontinuous first derivative, corresponding to a first order phase transition behavior, and plug it in that inverse formula getting some density of states g(E) as a result. What I'm interested in, is what kind of features in g(E) correspond to a phase transition.