The volume entropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls in its universal cover. This concept is closely related with other notions of entropy found in dynamical systems and plays an important role in differential geometry and geometric group theory. If the manifold is nonpositively curved then its volume entropy coincides with the topological entropy of the geodesic flow. It is of considerable interest in differential geometry to find the Riemannian metric on a given smooth manifold which minimizes the volume entropy, with locally symmetric spaces forming a basic class of examples.
The volumeentropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls...
Entropy is a scientific concept that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used...
S {\textstyle S} is the entropy of the system T {\textstyle T} is the temperature of the system V {\textstyle V} is the volume of the system p {\textstyle...
in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on δ E {\displaystyle \delta...
In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in...
capacity for entropy. This quantity is the amount of entropy that may be increased without changing an internal energy or increasing its volume. In other...
law of thermodynamics requires that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing...
In thermodynamics, entropy is a numerical quantity that shows that many physical processes can go in only one direction in time. For example, cream and...
absolute temperature, and S is the entropy. H = U + pV, where U is the internal energy, p is the pressure, and V is the volume. G is the most useful for processes...
discovered recently (see paper by Katz and Sabourau below) that the volumeentropy h, together with A. Katok's optimal inequality for h, is the "right"...
entropy that do not depend on ω - the volume and surface entropy described above. (Note that the surface entropy differs from the Boltzmann entropy only...
Research concerning the relationship between the thermodynamic quantity entropy and both the origin and evolution of life began around the turn of the...
The mathematical expressions for thermodynamic entropy in the statistical thermodynamics formulation established by Ludwig Boltzmann and J. Willard Gibbs...
Entropy production (or generation) is the amount of entropy which is produced during heat process to evaluate the efficiency of the process. Entropy is...
especially convenient for calculating changes in entropy since, in this case, the formula for the entropy change, ΔS, is simply Δ S = Q rev T {\displaystyle...
an entropic force acting in a system is an emergent phenomenon resulting from the entire system's statistical tendency to increase its entropy, rather...
the entropy is an exact differential, using the chain rule, the change in entropy when going from a reference state 0 to some other state with entropy S...
The third law of thermodynamics states that the entropy of a closed system at thermodynamic equilibrium approaches a constant value when its temperature...
principle states that the entropy of ordinary mass (not just black holes) is also proportional to surface area and not volume; that volume itself is illusory...
Entropy is one of the few quantities in the physical sciences that require a particular direction for time, sometimes called an arrow of time. As one...
would show that the initial and final entropies are the same, thus, the reason it is called isentropic (entropy does not change). Thermodynamic processes...
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