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I am trying to understand the basics of thermodynamics of dissipative systems.

In the attached paper equation (2.13) is derived using the concpt of free-energy and Onsager's relations

It says that, restricting to only one generalized coordinate

∂V / ∂q + ∂D / ∂q' = Q

V being the free energy, D the dissipation function as defined by Onsager's principle, Q the generalized force associated to the generalized coordinate q, and the apex deoting differentiation wit respect to time.

Trying to make sense, I applied to a one-dimensional dissipative sisyem, a dashpot.

Indeed, for a dashpot V = 0 identically, so the equation suggests (D defined as D = 0.5 b q'^2)

b q' = Q

which makes perfect sense (newtonian viscosity).

If I try to do the same with a spring and dashpot in series, having q1 and q2 as coordinates representing the extension of the spring and the dashpot, I end up with a system ,

∂V / ∂q_1 + ∂D / ∂q'_1 = Q_1

∂V / ∂q_2 + ∂D / ∂q'_2 = Q_2

The first makes perfect sense, expressing the fact the force in the spring equals the applied force.

The second should say the same for the dashpot, but I struggle to understand what Q_2 is. Should be a force conjugate to the displacement of the dashpot, but the dashpot react to an applied rate, not to an applied displacement.

Where am I going wrong?

Thanks

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# Dissipative Thermodynamics

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