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In this work, we study the effects of random temperature fluctuations on theequation of state of a non-interacting, relativistic fermion gas by means ofthe replica method. This picture provides a conceptual model for anon-equilibrium system, depicted as an ensemble of subsystems at differenttemperatures, randomly distributed with respect to a given mean value. We thenassume the temperature displays stochastic fluctuations $T = T_0 + \delta T$with respect to its ensemble average value $T_0$, with zero mean$\overline{\delta T} = 0$ and standard deviation $\overline{\delta T^2} =\Delta$. By means of the replica method, we obtain the average grand canonicalpotential, leading to the equation of state of the fermion gas expressed interms of the excess pressure caused by these fluctuations with respect to theideal gas at uniform temperature. We further extend our results for the idealBose gas as well. Our findings reveal an increase in pressure as the system'sensemble average temperature $T_0$ rises, consistently exceeding the pressureobserved in an equilibrium state. Finally, we explore the implications for thedeconfinement transition in the context of the simple Bag model, where we showthat the critical temperature decreases.

Temperature fluctuations in a relativistic gas: Pressure corrections and possible consequences in the deconfinement transition