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Can the derivative defining pressure $dU \over dV$ or ${∂S \over ∂V}|_{E,N} $ be negative in processes occuring in system not cosmological but statistical (gases or solids or liquids - I mean the statistical study of systems). Altough I have read about negative temperatures and negative pressures, could we have for a system at positive temperature, a negative price of pressure (absolute)?

Note: I have read Are negative temperatures typically associated with negative absolute pressures?

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Depending on how "real" you want the system to be. In a Casimir plate setup in a way the vacuum gets a negative pressure. But for any system of real particles a negative pressure means, that the system will be unstable and collapse, so you cannot have negative pressure in equilibrium. For example, negative pressure can formally occur in the van der Waals model of non-ideal gases, but there it is only an artifact as the uniform phase is unstable (even when the pressure does not drop below zero), and instead there is an equilibrium between gas and liquid at a constant temperature and pressure.

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    $\begingroup$ It is worth mentioning that the metastable state (before the collapse), while not an equilibrium state, can sometimes be maintained for very long times if the substance is extremely pure (similar to supercooled purified water), or if the system is very small so that nucleation centers don't take. In fact here it was argued that trees would not be able to grow as tall as they do were they not able to upkeep negative pressures. $\endgroup$
    – alarge
    Commented Jun 12, 2015 at 1:48
  • $\begingroup$ Hi, and thanks for the answer. Could you elaborate or give me a clue on how to understand why from negative pressure follows an unstable system that will collapse?Is there a way to see this statistically nad thermodynamically?Thank you. $\endgroup$ Commented Jun 13, 2015 at 8:17
  • $\begingroup$ @alarge Hi. I have read your comment and the answer you gave in the question you mention, and I see that as the answer above, the negative pressure is interpreteted as a situation in a metastable state. Thus, couldn't in any way one have a negative pressure in a stable state? Thanks. $\endgroup$ Commented Jun 13, 2015 at 8:26
  • $\begingroup$ @ConstantineBlack No, a negative pressure state cannot be an equilibrium state (because the entropy could be made higher by contracting the body, so this would happen spontaneously. The reason there is a metastability rather than a strict requirement for pressures to be positive is that contracting involves creating new surface area which means there would first be an increase in free energy before the eventual decrease). That said, if the state is stable enough, you can probably treat it as if it were a stable state and use the equations of thermodynamics to describe some secondary phenomena. $\endgroup$
    – alarge
    Commented Jun 13, 2015 at 11:48
  • $\begingroup$ This answer might be right for solids and liquids, but OP explicitly mentions solids, which can be under tension at equilibrium without collapsing. $\endgroup$ Commented Jul 6, 2019 at 6:45

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