Statistical mechanics allows us to consider an ensemble of systems, each of which consisting of only a single particle. Once we write the partition function for the system of one particle, we can easily derive all the thermodynamic quantities. One can accept that the internal energy computed from the partition function is the average energy of the system. But how to interpret pressure? What does pressure mean in the case of a system of one particle.

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Arjun Wright · Accepted Answer

Pressure is defined as the rate of increase in internal energy to rate of decrease in volume, i.e.  P  =  −            ∂      U              ∂      V      Assume a particle in a box, for example the classic infinite quantum potential well of width L. The quantized energy is      E    n    =                    n        2                    h        2                    8      m              L        2            In a 3D box this becomes      E                  n        x            ,              n        y            ,              n        z              =            (              n        x        2            +              n        y        2            +              n        z        2            )              h        2                    8      m              L        2              =            (              n        x        2            +              n        y        2            +              n        z        2            )              h        2                    8      m              V                  2                      /                    3                    where V is the volume.The ground state energy for this system is      E    0    =            3              h        2                    8      m              V                  2                      /                    3                    therefore the pressure is  P  =  −            ∂              E        0                    ∂      V        =            h      2              4      m              V                  5                      /                    2                    So you can see even a single particle can excert pressure on the boundaries of its bounding box!

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