Gamma vs. lognormal distributions I have an experimentally observed distribution that looks very similar to a gamma or lognormal distribution. I've read that the lognormal distribution is the maximum entropy probability distribution for a random variate $X$ for which the mean and variance of $\ln(X)$ are fixed. Does the gamma distribution have any similar properties?
 A: As for qualitative differences, the lognormal and gamma are, as you say, quite similar.
Indeed, in practice they're often used to model the same phenomena (some people will use a gamma where others use a lognormal). They are both, for example, constant-coefficient-of-variation models (the CV for the lognormal is $\sqrt{e^{\sigma^2} -1}$, for the gamma it's $1/\sqrt \alpha$).
[How can it be constant if it depends on a parameter, you ask? It applies when you model the scale (location for the log scale); for the lognormal, the $\mu$ parameter acts as the log of a scale parameter, while for the gamma, the scale is the parameter that isn't the shape parameter (or its reciprocal if you use the shape-rate parameterization). I'll call the scale parameter for the gamma distribution $\beta$. Gamma GLMs model the mean ($\mu=\alpha\beta$) while holding $\alpha$ constant; in that case $\mu$ is also a scale parameter. A model with varying $\mu$ and constant $\alpha$ or $\sigma$ respectively will have constant CV.]
You might find it instructive to look at the density of their logs, which often shows a very clear difference.
The log of a lognormal random variable is ... normal. It's symmetric.
The log of a gamma random variable is left-skew. Depending on the value of the shape parameter, it may be quite skew or nearly symmetric.
Here's an example, with both lognormal and gamma having mean 1 and variance 1/4. The top plot shows the densities (gamma in green, lognormal in blue), and the lower one shows the densities of the logs:

(Plotting the log of the density of the logs is also useful. That is, taking a log-scale on the y-axis above)
This difference implies that the gamma has more of a tail on the left, and less of a tail on the right; the far right tail of the lognormal is heavier and its left tail lighter. And indeed, if you look at the skewness, of the lognormal and gamma, for a given coefficient of variation, the lognormal is more right skew ($\text{CV}^3+3\text{CV}$) than the gamma ($2\text{CV}$).
A: Yes, the gamma distribution is the maximum entropy distribution for which the mean $E(X)$ and mean-log $E(\log X)$ are fixed.  As with all exponential family distributions, it is the unique maximum entropy distribution for a fixed expected sufficient statistic.
To answer your question about physical processes that generate these distributions:  The lognormal distribution arises when the logarithm of X is normally distributed, for example, if X is the product of very many small factors.  If X is gamma distributed, it is the sum of many exponentially-distributed variates.  For example, the waiting time for many events of a Poisson process.
