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It has long been known that there is no measurable heat production associated with the nerve pulse. Rather, one finds that heat production is biphasic, and a heat release during the first phase of the action potential is followed by the reabsorption of a similar amount of heat during the second phase. We review the long history the measurement of heat production in nerves and provide a new analysis of these findings focusing on the thermodynamics of adiabatic and isentropic processes. We begin by considering adiabatic oscillations in gases, waves in layers, oscillations of springs and the reversible (or irreversible) charging and discharging of capacitors. We then apply these ideas to the heat signature of nerve pulses. Finally, we compare the temperature changes expected from the Hodgkin-Huxley model and the soliton theory for nerves. We demonstrate that heat production in nerves cannot be explained as an irreversible charging and discharging of a membrane capacitor as it is proposed in the Hodgkin-Huxley model. Instead, we conclude that it is consistent with an adiabatic pulse. However, if the nerve pulse is adiabatic, completely different physics is required to explain its features. Membrane processes must then be reversible and resemble the oscillation of springs more than resembling "a burning fuse of gunpowder" (quote A. L. Hodgkin). Theories acknowledging the adiabatic nature of the nerve pulse have recently been discussed by various authors. It forms the central core of the soliton model, which considers the nerve pulse as a localized sound pulse.