Question

Integrals over the Fermi–Dirac and Bose–Einstein distributions define a “poly” form of this function. The replica trick takes advantage of the fact that this function (10[1])of Z is equal to the limit as n approaches zero of “Z-to-the-n, minus one, all over n.” The expected value of the energy is equal to the negative partial derivative of this function of Z with respect to beta. The Helmholtz free energy is equal to kT times this function of the partition function. Stirling’s formula approximates (-5[1])this function of n-factorial. (10[1])The negative (10[1])sum over all states of “p times this function of p” defines the Shannon entropy, while Boltzmann’s definition of entropy is equal (10[1])to k times this function of the number of microstates. For 10 points, name this function, the inverse of the exponential function. (10[1])■END■ (10[4]0[1])

ANSWER: natural logarithm [or natural log or ln; accept polylogarithm]
<Physics>
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Rasheeq Azad (UG)UNC B (UG)Duke A (UG)2410
Kenny Zhang (UG)Virginia A (UG)GWU B (Grad)82-5
Jim Fan (Grad)UNC A (Grad)Maryland B (UG)8610
Isaac Mammel (UG)Maryland A (Grad)JMU B (UG)8810
Brian Lai (DII)Virginia B (UG)UNC D (DII)11010
Perry O'Connor (Grad)Liberty A (Grad)Liberty C (DII)13210
Kevin Liu (DII)Maryland C (DII)GWU A (UG)13310
Carly Hemani (DII)JMU A (UG)Roanoke College A (DII)13310
Michael Eng (UG)UNC C (UG)Liberty B (DII)13310
Annie Goodman (Grad)GWU B (Grad)Virginia A (UG)1330
Zander Werner (DII)Virginia C (UG)William & Mary A (UG)13310