A Dependence of entropy and temperature on Planck's constant

1. Apr 20, 2017

spaghetti3451

Consider the dependence of entropy and of temperature on the reduced Planck's constant (taken from page 23 of Thomas Hartman's lecture notes(http://www.hartmanhep.net/topics2015/) on Quantum Gravity):

$$S \propto \hbar, \qquad \qquad T \propto \hbar.$$

I do not quite see how entropy can depend on the reduced Planck's constant. To give credence to my claim, consider the definition of (statistical) entropy in classical and in quantum systems.

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In a classical system, the (statistical Gibbs) entropy for a macroscopic system (with a discrete set of microstates) is defined as

$$S = -k_\text{B}\,\sum_i p_i \ln \,p_i,$$

where $k_{\text{B}}$ is Boltzmann's constant and $p_i$ is the probability that the system is in microstate $i$ during the system's fluctuations.

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In a quantum system, the (statistical von Neumann) entropy for a macroscopic system (with a discrete set of microstates) is defined as

$$S = -k_\text{B}\,\text{Tr}\ (\rho \ln \rho),$$

where $k_{\text{B}}$ is Boltzmann's constant and $\rho$ is the density matrix of the system.

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How then does entropy depend on the reduced Planck's constant in quantum systems?

Why should temperature also depend on the reduced Planck's constant in quantum systems?

2. Apr 21, 2017

Jilang

I can see how the number of distinguishable miscrstates might be related to the inverse of h, but not directly proportional to it. (As h tends to zero the number of micro states would to infinity.)