On ruelle's construction of the thermodynamic limit for the classical microcanonical entropy

TY - JOUR

T1 - On ruelle's construction of the thermodynamic limit for the classical microcanonical entropy

AU - Kiessling, Michael K.H.

N1 - Funding Information: Acknowledgement I thank Sheldon Goldstein, Joel L. Lebowitz, and the two referees for their helpful comments on the manuscript. Support by the NSF under Grant DMS-0406951 is gratefully acknowledged.

PY - 2009/1

Y1 - 2009/1

N2 - In 1969 Ruelle published his construction of the thermodynamic limit, in the sense of Fisher, for the quasi-microcanonical entropy density of classical Hamiltonian N-body systems with stable and tempered pair interactions. Here, "quasi-microcanonical" refers to the fact that he discussed the entropy defined with a regularized microcanonical measure as ln∈(N! -1 χ {-?<H<}d6N X) rather than defined with the proper microcanonical measure as ln∈(N!-1 δ(-H)∈d6N X). Replacing δ(-H) by χ {-?<H<} seems to have become the standard procedure for rigorous treatments of the microcanonical ensemble hence. In this note we make a very elementary technical observation to the effect that Ruelle's proof (still based on regularization) does establish the thermodynamic limit also for the entropy density defined with the proper microcanonical measure. We also show that with only minor changes in the proof the regularization of δ(-H) is actually not needed at all.

AB - In 1969 Ruelle published his construction of the thermodynamic limit, in the sense of Fisher, for the quasi-microcanonical entropy density of classical Hamiltonian N-body systems with stable and tempered pair interactions. Here, "quasi-microcanonical" refers to the fact that he discussed the entropy defined with a regularized microcanonical measure as ln∈(N! -1 χ {-?<H<}d6N X) rather than defined with the proper microcanonical measure as ln∈(N!-1 δ(-H)∈d6N X). Replacing δ(-H) by χ {-?<H<} seems to have become the standard procedure for rigorous treatments of the microcanonical ensemble hence. In this note we make a very elementary technical observation to the effect that Ruelle's proof (still based on regularization) does establish the thermodynamic limit also for the entropy density defined with the proper microcanonical measure. We also show that with only minor changes in the proof the regularization of δ(-H) is actually not needed at all.

KW - Classical microcanonical entropy

KW - Thermodynamic limit

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U2 - https://doi.org/10.1007/s10955-008-9638-4

DO - https://doi.org/10.1007/s10955-008-9638-4

M3 - Article

SN - 0022-4715

VL - 134

SP - 19

EP - 25

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 1

ER -