In this section we consider the constraints from demanding consistency with an equilibrium partition function. As this calculation is not that relevant for our work in the following chapter, we just provide a schematic outline for the sake of completeness.
The fluid stress tensor comprises of dissipative and non-dissipative terms:
Tµν =Tµνdisp+Tµνnon−disp. (5.10)
Each term from first order onwards will have an attached transport coefficient. The dissipative terms will vanish when evaluated on equilibrium configurations and thus cannot be constrained by equilibrium considerations. Demanding consistency with an equilibrium partition function will only impose constraints on the transport coefficients arising from Tµνnon−disp.
Our constraints from this principle are derived as follows. We first parametrise all possible equilibrium solutions. For a fluid on given stationary spacetime background gAB(xi) (wherexi represent spatial coordinates only) the fluid variables will necessarily be functions of this metric (there are no other characteristic scales in the system). For equilibrium configurations, the velocity and temperature field, ueqµ and Teq must take the following functional form:
ueqµ =uµ(gAB(xi))
Teq =T(gAB(xi)). (5.11)
As a consequence, the stress tensor will evaluate to:
Tµνeq =Tµνnon−disp(gAB(xi)). (5.12) The result above gives us a set of expressions for the stress tensor terms at each order in the derivative expansion.
Another approach to evaluating the stress tensor on equilibrium configurations would be to construct the partition function; again, this must be a function of the background metric fields:
logZ = logZ(gAB(xi)). (5.13)
The equilibrium stress tensor would then be proportional to this derivative of the partition function:
Tµνpartition ∝ δlogZ
δgAB . (5.14)
Equating these two expressionsTµνpartitionandTµνeq will give us equality constraints for the non-dissipative transport coefficients at each order in the derivative expansion.
Note that for an uncharged fluid at first order, the stress tensor only has two transport coefficients η and ζ, the shear and bulk viscosities, which are both dissipative terms.
Thus there can be no equality constraints at first order if we restrict to the uncharged case. This matches what we found in the previous section when we derived the constraints using the principle of entropy increase.
Stability of equilibrium in fluid dynamics
6.1 Introduction
In the previous chapter, we considered two physical principles which led to sets of constraints on the transport coefficients. The first was that the divergence of the fluid dynamical entropy current must be non-negative for all admissible fluid flows. This is a local form of the second law of thermodynamics. The second physical principle that we examined was compatibility with an equilibrium partition function. In general, the equality-type relations which result from the second law exactly match the constraints imposed by compatibility with an equilibrium partition function. The requirement of entropy increase also yielded a set of inequalities; these were not obtained by the equilibrium partition function method.
It is very interesting to consider how exactly these two physical principles are related.
For example, if we could recast the second law of thermodynamics in the language of partition functions, it would make its microscopic origins much clearer. As things stand, these two physical principles are not completely equivalent given the additional inequalities that result from the second law. This naturally motivates the question:
What further ingredient do we need to add for our equilibrium considerations to yield exactly the same set of constraints as the second law? It was conjectured in [39] that compatibility with an equilibrium partition function together with the requirement
that the equilibrium solutions be dynamically stable will be equivalent to positivity of the divergence of the entropy current.
We now pause to consider this conjecture further. It is well-known that the second law of thermodynamics implies stability of equilibrium. This is intuitively clear.
The equilibrium configuration would be at a local maximum of the entropy; small perturbations away from this would then evolve back to the same equilibrium given that entropy is constrained to always increase. This conjecture, however, concerns the reverse implication. Can the existence of equilibrium together with the requirement of stability imply the second law for fluid dynamics? This would be an immensely significant statement. It would imply that statements concerning small perturbations around equilibrium could result in the same constraints as the second law which holds for all non-equilibrium fluid flow, however far from equilibrium.
In this chapter, we consider this conjecture in the simplest possible case. We study uncharged fluid dynamics on a flat spacetime background to first order in the derivative expansion. A linear stability analysis is done and we determine the constraints imposed by stability about equilibrium. It turns out that the inequalities that we obtain are slightly weaker than the inequalities imposed by the second law.
We now outline the structure of this chapter. In section 6.2, we consider small amplitude perturbations about equilibrium for linearised hydrodynamics in flat spacetime; the requirement of linear stability imposes a set of inequalities which we list here. Reasons for the inequivalence between the two sets of inequalities derived from stability of equilibrium and the second law of thermodynamics are then discussed in section 6.3.
We end with our conclusions in the final section 6.4.