In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless, it is a pure number.
Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for generalizations. The partition function has many physical meanings, as discussed in Meaning and significance.
Is it possible to obtain the relation
S = \log Z + \langle U \rangle /T
directly from the Boltzmann distribution?
Edit: It seems that we can if we use the VN entropy:
S = -\Sigma p_i \log p_i
This suggests that the entropy of a single microstate should be
s = -\log( \frac{e^{-\epsilon...
Partition function for a gas in a cylinder -- urgent!
Hi,
Here's the problem -- it's supposed to be a specimen of what I can expect in my exam, but it isn't much like the tutorial questions I've been doing. I'd really appreciate some help -- fast!
Homework Statement
An ideal gas consisting...
Is the energy given by the first or the second? I have seen both relationships in different websites, and I am confused.
E = kT^2 \frac{\partial Z}{\partial T}
or
E = - \frac{\partial ln Z}{\partial \beta}
The partition function in the classical theory is an integral over phase space. Thus, the partition function is often not dimensionless. Then the formula
F = -T \log Z
can no longer be valid, as you can only take the logarithm of a dimensionless number. In the quantum theory, this...
What is the relation between the partition function and the Boltzman, Maxwell distribution?
Differences and similarities?
Both have exponentials to the power of the negative total energy of the microstate. Although the word microstate dosen't occur in the Boltzman, Maxwell case.
Is the BM...
Homework Statement
When modeling ideal gas molecules using a grand partition ensemble, is heat flow = 0? So if U=Q-W then in a grand canonical ensemble, U=-W?The Attempt at a Solution
I think so as the system is in thermal equilibrium with the surroundings. So in this system the total energy is...
Let us consider a collection of non-interacting hydrogen atoms at a certain temperature T.
The energy levels of the hydrogen atom and their degeneracy are:
En = -R/n²
gn = n²
The partition function in statistical physics is given by:
Z = Sum(gn Exp(-En/kT), n=1 to Inf)
This...
If I have a 2 state system with energy levels of the 2 states to be 0 and V. I find the partition function to be Z = 1 + e^(-V/kT). Am I correct? If so, does that not mean the average energy is V? and thus the entropy is 0? This doesn't make sense, how is the entropy of a 2 state system (when 1...
Let be the continuous partition function:
Z(\beta)=(N!)^{-1}\int_{V}dx_1 dx_2 dx_3 dx_4 ...dx_N exp(-\beta H(x_1, x_2 , x_3 , ... ,x_n,p_1 , p_2 , ..., p_N
if the Hamiltonian is 'quadratic' in p's are q's do this mean that the particles in the gas solid or whatever follow a Normal...
Let be the Hamiltonian Energy equation:
H\Psi= E_{n} \Psi
then let be the partition function:
Z=\sum_{n} g(n)e^{-\beta E_{n}}
where the "Beta" parameter is 1/KT k= Boltzmann constant..the question is..let,s suppose we know the "shape" of the function Z...could we then "estimate"...
This is a question about thermal physics. There's this partition function Z = sum over all states s of the system ( exp(-E_s/T)). And its just used to calculate the probability of any state by taking the Boltzman factor exp(-E_s/T) of that state and dividing over the partition function. Theres...
I got a problem by finding an proper explanation.
The Boltzmann factor is defined as
P_j=\frac{1}{Z}e^{-\beta E_j}
I know, this is a probability distribution. but what exactly does it mean?
Wikipedia says: "The probability Pj that the system occupies microstate j" (link)
But that doesen...
Hi,
i'm having trouble with a thermal physics problem relating to the partition function and i was wondering if anyone could help me out. the problem is as follows:
(a) Consider a molecule which has energy levels En=c|n| , where n is a vector with integer components. Compute the partition...