### The Legendre Transform of the Cumulant Generating Function

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Here are two references regarding the mathematical meaning of the ubiquitous “effective action”:

* H. Touchette, “The large deviation approach to statistical mechanics” [arXiv:0804.0327]

* Shalizi, C. “Large Deviations for IID Sequences: The Return of Relative Entropy”, in “Almost None of Stochastic Processes”, lecture notes (2006) [PDF].

To wit, the way I found these was to look for “Legendre transform cumulant generating function”, because this is what the effective action obviously really is. For the people not familiar from statistics with the concept of “cumulants”, here is a MO post where it's explained a bit more (which I find slightly odd, because it's actually explained with physics buzzwords, like "interaction", which in turn I would like to avoid just by using probabilistic terminology).

The second claims that the effective action is the relative entropy, which is intriguing.

To quote from the first,

“The mathematical theory of large deviations initiated by Cramér in the 1930s, and later developed by Donsker and Varadhan and by Freidlin and Wentzell in the 1970s, is not a theory commonly studied in physics. Yet it could be argued, without being paradoxical, that physicists have been using this theory for more than a hundred years, and are even responsible for writing down the very first large deviation result. Whenever physicists calculate an entropy function or a free energy function, large deviation theory is at play. In fact, large deviation theory is almost always involved when one studies the properties of many-particle systems, be they equilibrium or nonequilibrium systems. So what are large deviations, and what is the theory that studies these deviations?”

EDIT:

File attached: quantumldtsources1.pdf (106.3 KB)

* H. Touchette, “The large deviation approach to statistical mechanics” [arXiv:0804.0327]

* Shalizi, C. “Large Deviations for IID Sequences: The Return of Relative Entropy”, in “Almost None of Stochastic Processes”, lecture notes (2006) [PDF].

To wit, the way I found these was to look for “Legendre transform cumulant generating function”, because this is what the effective action obviously really is. For the people not familiar from statistics with the concept of “cumulants”, here is a MO post where it's explained a bit more (which I find slightly odd, because it's actually explained with physics buzzwords, like "interaction", which in turn I would like to avoid just by using probabilistic terminology).

The second claims that the effective action is the relative entropy, which is intriguing.

To quote from the first,

“The mathematical theory of large deviations initiated by Cramér in the 1930s, and later developed by Donsker and Varadhan and by Freidlin and Wentzell in the 1970s, is not a theory commonly studied in physics. Yet it could be argued, without being paradoxical, that physicists have been using this theory for more than a hundred years, and are even responsible for writing down the very first large deviation result. Whenever physicists calculate an entropy function or a free energy function, large deviation theory is at play. In fact, large deviation theory is almost always involved when one studies the properties of many-particle systems, be they equilibrium or nonequilibrium systems. So what are large deviations, and what is the theory that studies these deviations?”

EDIT:

**List of references on quantum large deviations**File attached: quantumldtsources1.pdf (106.3 KB)

Here is another tutorial on large deviations that I found.

written
11 weeks ago by
Tim