Tag Info

Hot answers tagged

2

We're used to thinking of "most probable" and "mean value" as the same thing, but it need not be so. It's worth remembering that the "expectation value" of a six sided die is 3.5, but this is not a very probable result. You might object that this is due to discrete effects, but consider this example: you have two identical Gaussians, with width $\sigma$, but ...


2

The intuitive picture is that phase points move more quickly through regions of phase space where $|\nabla H|$ is higher. As a result if you have a constant-energy ensemble of phase points (a flat "packet"), their phase space area enlarges as they move through high $|\nabla H|$ regions. For the full Liouville's theorem this is not a problem. A ...


1

I think the wrong step was the assumption that entropy increases, where in fact maintaining the temperature would require a an outflow of heat, which means the entropy of the gas is decreasing. To see how this relates to your formula, notice that this decrease in entropy would also increase the enthalpy by the same amount, however enthalpy will also ...


1

The wave function for a pair of independent, indistinguishable bosons/fermions with no internal degrees of freedom can be written in the position representation as $$ \Psi(r_1,r_2) = \frac{1}{\sqrt{2}} \left[\psi_1(r_1)\psi_2(r_2) \pm \psi_2(r_1)\psi_1(r_2) \right ].$$ You can calculate your desired probability distribution from $P_{12}(r_1,r_2) = ...


1

I don't know if this is standard, but consider a pendulum that can swing a full circle in a plane. Vibrate the point of suspension up and down at the appropriate frequency. The pendulum will gain energy and spin either clockwise or counterclockwise.


1

In any probability distribution, there are many ways to find some kind of "average" value, that is, ways to define the "centrality" of the distribution. In discreet distributions you have almost certainly come across mean, median and mode, and perhaps also the different "flavours" of means - arithmetic, geometric, harmonic etc. For continuous distributions ...


1

Your requirement that $z\to 1$ (i.e. $\mu\to 0$) already requires the limit $N\to \infty$. If $N$ is finite, the chemical potential tends to a small but finite value at the transition temperature. This ensures that the occupation of the ground state is $\langle n_0\rangle \lesssim N$.


1

If we consider temperature to be due to translational motion of the molecules and we assume the system has reached equilibrium, then the velocity distribution of the molecules is given by the Maxwell distribution: $$ f(v) = \sqrt{\left(\frac{m}{2\pi k T}\right)^3} 4 \pi v^2 \exp\left(\frac{m v^2}{2 k T}\right)$$ which will give you the velocity ...


1

Correlation between two variables (or objects) is, very simply put, how much a change in one variable affects or determines a change in the other. Replacing variables with spins, highly correlated spins would mean that, due to some interactions between them, a change in the direction of one spin will cause a change in the direction of the spin it is ...


1

One way to understand it is to write it using the Dirac measure to express the phase space in the microcanonical ensemble (because that's what it is about). In this ensemble the idea is to say that the energy $H(\textbf{r})$ (where $r$ is a point in phase space) is fixed at some value $E$ (actually it belongs to a very small interval $[E,E+\delta E]$). One ...



Only top voted, non community-wiki answers of a minimum length are eligible