Free and Classical Entropy over the Circle
نویسنده
چکیده
Relative entropy with respect to normalized arclength on the circle is greater than or equal to the negative logarithmic energy (Voiculescu’s negative free entropy) and is greater than or equal to the modified relative free entropy. This note contains proofs of these inequalities and related consequences of the first Lebedev–Milin inequality.
منابع مشابه
Rényi entropies of free bosons on the torus and holography
We analytically evaluate the Rényi entropies for the two dimensional free boson CFT. The CFT is considered to be compactified on a circle and at finite temperature. The Rényi entropies Sn are evaluated for a single interval using the two point function of bosonic twist fields on a torus. For the case of the compact boson, the sum over the classical saddle points results in the Riemann-Siegel th...
متن کاملOn Classical Analogues of Free Entropy Dimension
Abstract. We define a classical probability analog of Voiculescu’s free entropy dimension that we shall call the classical probability entropy dimension. We show that the classical probability entropy dimension is related with diverse other notions of dimension. First, it equals the fractal dimension. Second, if one extends Bochner’s inequalities to a measure by requiring that microstates aroun...
متن کامل`2–torsion of Free-by-cyclic Groups
We provide an upper bound on the `2–torsion of a free-bycyclic group, −ρ(2)(F oΦ Z), in terms of a relative train-track representative for Φ ∈ Aut(F). Our result shares features with a theorem of Lück–Schick computing the `2–torsion of the fundamental group of a 3–manifold that bers over the circle in that it shows that the `2–torsion is determined by the exponential dynamics of the monodromy. ...
متن کاملA TAXICAB VERSION OF A TRIANGLE' S APOLLONIUS CIRCLE
One of the most famous problems of classical geometry is the Apollonius' problem asks construction of a circle which is tangent to three given objects. These objects are usually taken as points, lines, and circles. This well known problem was posed by Apollonius of Perga ( about 262 - 190 B.C.) who was a Greek mathematician known as the great geometer of ancient times after Euclid and Archimede...
متن کاملIvestigation of Entropy Generation in 3-D Laminar Forced Convection Flow over a Backward Facing Step with Bleeding
A numerical investigation of entropy generation in laminar forced convection of gas flow over a backward facing step in a horizontal duct under bleeding condition is presented. For calculation of entropy generation from the second law of thermodynamics in a forced convection flow, the velocity and temperature distributions are primary needed. For this purpose, the three-dimensional Cartesian co...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006