Isothermal axisymmetric equilibria of gravitating rods are generally stable with respect to cylindrically-symmetric perturbations. In particular, there is no gravothermal catastrophe as in isothermal spheres. But no isothermal equilibria exist below some finite temperature and, in contact with a heat bath slightly colder than that, the system collapses slowly, giving up an unlimited amount of energy.
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Katz et al. (1978) studied this question.
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