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We calculate the thermodynamic properties of a collection of $N$ smallJosephson junctions coupled to a single-mode resonant electromagnetic cavity,at finite temperature $T$, using several approaches. In the first approach, weinclude all the quantum-mechanical levels of the junction, but treat thejunction-cavity interaction using a mean-field approximation developedpreviously for $T = 0$. In the other approaches, the junctions are treatedincluding only the two lowest energy levels per junction, but with twodifferent Hamiltonians. The first of these maps onto the Dicke model of quantumoptics. The second is a modified Dicke model which contains an additionalXY-like coupling between the junctions. The modified Dicke model can be treatedusing a mean-field theory, which in the limit of zero XY coupling gives thesolution of the Dicke model in the thermodynamic limit using Glauber coherentstates to represent the cavity. In all cases, for an $N$-independentjunction-cavity coupling, there is a critical junction number $N$ above whichthere is a continuous transition from incoherence to coherence with decreasing$T$. If the coupling scales with $N$ so as to give a well-behaved thermodynamiclimit, there is a critical minimum coupling strength for the onset ofcoherence. In all three models, the cavity photon occupation numbers have anon-Bose distribution when the system is coherent.

Coherence Transition of Small Josephson Junctions Coupled to a Single-Mode Resonant Cavity: Connection to the Dicke Model