The analogues of entropy and of Fisher's information measure in free probability theory. II
In Part I [Commun. Math. Phys. 155, No. 1, 71-92 (1993; Zbl 0781.60006)] we studied the free entropy \(\Sigma(X)\) of a self-adjoint random variable \(X\) equal to minus the logarithmic energy of its distribution. We also introduced a generalization \(\Sigma(X_ 1,\dots, X_ n)\) which, however, does not have the necessary properties for \(n \geq 2\) to represent the joint free entropy of \(X_ 1, \dots, X_ n\) though it has another natural interpretation. Here we take another route based on the volumes of sets of matrix approximants. This can be interpreted as a normalized limit of the logarithm of the statistical weight of approximating microstates -- i.e. is in tune with Boltzmann's original point of view. We obtain a quantity \(\chi (X_ 1, \dots, X_ n)\) which appears to have the right properties. For \(n = 1\), \(\chi(X)\) and \(\Sigma(X)\) coincide (up to an additive constant). We will therefore call \(\chi(X_ 1, \dots, X_ n)\) the free entropy of \((X_ 1, \dots, X_ n)\) (we will call \(\Sigma(X_ 1,\dots, X_ n)\) the orbital weight). For free random variables, \(\chi(X_ 1, \dots, X_ n) = \chi(X_ 1) + \cdots + \chi(X_ n)\). Moreover, \(\chi(X_ 1, \dots, X_ n)\) has the right transformation properties with respect to noncommutative functional calculus. We also introduce another quantity: the free entropy dimension \(\delta(X_ 1,\dots,X_ n)\). The relation of \(\delta\) to \(\chi\) is similar to the relation in geometric measure theory of Minkowski dimension to Lebesgue measure. For an \(n\)-tuple of free selfadjoint random variables we explicitly compute \(\delta (X_ 1,\dots, X_ n)\) and it is interesting to note that this coincides, in case they generate a \(\Pi_ 1\)-factor with its ``free dimension'' parameter in [\textit{K. Dykema}, Duke Math. J. 69, No. 1, 97-119 (1993; Zbl 0784.46044)]. Another result we obtain is, roughly stated, that: If a \(\Pi_ 1\) factor is generated by \(n\) self-adjoint elements \(X_ 1, \dots, X_ n\) and there is also a semicircular system of self-adjoint generators \(Y_ 1, \dots, Y_ m\) which are ``smooth noncommutative functions of \(X_ 1, \dots, X_ n\)'', then \(n \geq m\). Of course, replacing ``smooth'' by ``\(L^ \infty\)'' in the above statement corresponds to the isomorphism problem for the free group factors.
statistical weight of approximating microstates
scientific article; zbMATH DE number 703862