I see this term coming up quite often, especially in the context of mathematical physics, but I cannot find it defined anywhere. What constitutes a physical measure and what are some of its properties?
EDIT: I saw this term on a few posts on the web, as well as a few papers I was reading. I will try to find these to provide some context, but in the meantime here is a SE post and a paper from arXiv that use the term:
What is the probability density function on solutions to the Lorenz system?
https://arxiv.org/pdf/1012.0513.pdf (this paper does define the term, but I was wondering if this is the general definition)
The notion of a physical measure seems to go back to Eckmann & Ruelle's paper "Ergodic theory of chaos and strange attractors" (p.626). A closely related notion, "SRB measure" also seems to go back to this paper, and in the model scenario indeed both of these are unique and they coincide (however there are SRB measures that are not physical measures (https://mathoverflow.net/q/409755/66883) and there are physical measures that are not SRB measures (https://mathoverflow.net/a/164683/66883)). Instead of using them interchangeably I'll follow Eckmann & Ruelle (as opposed to following Viana). (Also relevant to this discussion is Young's paper "What are SRB Measures and Which Dynamical Systems Have Them?", in particular p.741).
(SRB stands for Sinai-Ruelle-Bowen; each of these mathematicians established existence and uniqueness of invariant measures that are "SRB" (whatever this means) for certain dynamical systems. On the other hand, the idea behind physical measures probably goes at least back to Oxtoby.)
First some preliminaries. If $X$ is a compact metrizable space and $f:X\to X$ is a homeomorphism, then the classical Krylov-Bogoliubov Theorem says that for any Borel probability measure $\nu$ any vague (= weakstar) limit point $\mu$ of the sequence of probability measures $\frac{1}{n}\sum_{k=0}^{n-1}f_\ast^k(\nu)$ gives an $f$-invariant Borel probability measure. (Here for $\phi:X\to \mathbb{R}$ a function by definition $\int_X \phi(x)\, d \frac{1}{n}\sum_{k=0}^{n-1}f_\ast^k(\nu)(x) = \frac{1}{n}\sum_{k=0}^{n-1}\int_X \phi\circ f^k(x)\, d\nu(x)$). In particular one could pick $\nu=\delta_x$ for some point $x\in X$, so that $\int_X \phi(y)\, d \frac{1}{n}\sum_{k=0}^{n-1}f_\ast^k(\delta_x)(y)=\frac{1}{n}\sum_{k=0}^{n-1}\phi\circ f^k(x)$ would be the time average of the values of $\phi$ along the trajectory of $x$ until time $n$.
Thus under quite broad assumptions there are invariant measures. Then the natural follow-up question would be to single out some invariant measures as special, as far as dynamics is concerned.
There are multiple ways in which one can do this, and one way is a claim of "physicality". Both the Ruelle-Eckmann paper and the Young paper mentions two such notions of physicality for measures, but typically only one of these notions is meant when the adjective "physical" is used (whereas the other is referred to as the "zero noise limit", and is attributed to Kolmogorov). Accordingly I'll focus further on this.
The main heuristic behind calling a measure "physical" is that Lebesgue measure (class) is the measure (class) that fits best to one's intuition of how big subsets are. In particular, one can pick points from sets of positive Lebesgue measure. Recall that while Lebesgue measure is defined on a Euclidean space, there is a related Lebesgue measure class on any $C^1$ manifold (this is simply the class of measures which locally are given by Lebesgue measure up to locally integrable densities that are bounded away from zero and infinity in local coordinates). One can thus safely talk about Lebesgue-negligible and Lebesgue-nonnegligible subsets of manifolds (alternatively one can fix a Riemannian metric, or a density to fix a measure in the Lebesgue measure class).
We are now ready to give the definition of a physical measure:
Def: Let $M$ be a compact $C^\infty$ manifold, $f:M\to M$ be a homeomorphism. Then an $f$-invariant Borel probability measure $\mu$ is said to be physical if the following subset of $M$ is not Lebesgue-negligible:
$$\mathbb{B}(\mu,f)=\left\{x\in M\,\left|\, \text{vaguelim}_{n\to\infty}\dfrac{1}{n}\sum_{k=0}^{n-1}f_\ast^k(\delta_x)=\mu\right.\right\}.$$
(Note that this definition is exactly the same definition runway44 wrote in a comment above, which is also the definition Viana & Yang use in their paper.)
Thus based on the heuristic above, physical measures are those measures that one can approximate well by choosing a random point "with one's eye" and computing the time averages along its trajectory.
Note that even though invariant measures are always guaranteed, physical measures may fail to exist, and even if they exist they may fail to be unique. See the papers linked above and references therein for theorems that guarantee both existence and uniqueness.