Effective Temperature

A few rambling thoughts before we begin: I've been feeling increasingly weak lately^[1], but after exercising today I feel refreshed and clear-headed, and my thinking has become noticeably more active. It seems like a good time to sort out a physics concept that has been bothering me recently (my memory really is poor, so this is as much for my future self as anyone else).

Introduction

The effective temperature discussed here refers to a concept that arises in the non-equilibrium states of ordinary classical systems: it allows certain properties of a non-equilibrium system to be mapped onto the same properties of an equilibrium system at a higher temperature.

From the fundamental laws of thermodynamics, the Gibbs relation, the FDR, kinetic energy, and potential energy, one can compute a "temperature" for a non-equilibrium system, called the effective temperature; different approaches yield different effective temperatures^[2]. The fact that different methods give inconsistent effective temperatures makes the concept elusive.^[3]

Vaguely, the effective temperature seems to be just a parameter, and one without a clear, universal meaning — even though the motion of a non-equilibrium system is unlikely to be the same in every direction and at every scale.

I would venture that at the end of the last century^[4] the concept of effective temperature was not very clear, but it can broadly be divided into two types: those computed from dynamical quantities and those computed from thermodynamic quantities. These days, fluctuation-dissipation relations (FDRs) are widely used. Who first computed it from FDRs remains to be verified, but Professor Ludovic Berthier has done a considerable amount of research in this area. This article will more or less center on the FDRs method throughout.

FDRs Method

In a non-equilibrium system under steady-state simple shear, for any two physical quantities $A\left(t\right)$ and $B\left(t\right)$, one can write their correlation (cross-correlation) function:

$$C_{AB}\left(t\right)=⟨A(t+t_0)B\left(t_0\right)⟩-⟨A\left(t_0\right)⟩⟨B\left(t_0\right)⟩$$

The response function:

$$R_{AB}\left(t\right)\equiv \frac{\delta A(t+t_0)}{\delta h_B\left(t_0\right)}$$

Here $h_B\left(t_0\right)$ is a perturbing field (acting on some of the particles) at time $t_0$ that is conjugate to $B(t)$. If one takes $A\left(t\right)=⟨\frac{1}{N^{\prime }}\sum _j{ϵ}_i\exp (i\mathbf{k}{\mathbf{r}}_{\mathbf{i}}\mathbf{(}\mathbf{t}\mathbf{)})⟩$ and $B(t)=⟨2\sum _j{ϵ}_j\cos (i\mathbf{k}{\mathbf{r}}_{\mathbf{i}}\mathbf{(}\mathbf{t}\mathbf{)})⟩$, then $C_{AB}(t)$ can be shown to be the Self-Intermediate Scattering Function (ISF), a quantity generally used to characterize relaxation among particles.

In equilibrium, the correlation function and the response function satisfy the fluctuation-dissipation theorem:

$$T{R}_{AB}(t)=\frac{d{C}_{AB}(t)}{dt}$$

Meanwhile, the susceptibility (I could not find a Chinese translation for this, so let us call it the "sensitization rate" for now) can be obtained by integrating the response function:

$${\chi }_{AB}(t)\equiv {\int }_0^{t}d{t}^{\prime }{R}_{AB}(t^{\prime })$$

If the field h_{B} is not too large and linear response holds, then the susceptibility can be used to write

$$T{\chi }_{AB}(t)=C_{AB}(0)-C_{AB}(t)$$

The effective temperature $T_{eff}^{AB}$ satisfies $R_{AB}(t)=-\frac{1}{T_{eff}^{AB}}\frac{d{C}_{AB}(t)}{dt}$, and on a plot of ${\chi }_{AB}-C_{AB}$ the effective temperature appears as the linear relation of the curve, as shown in the figure:

One glance at this figure makes it clear that there are a great many tricks involved in computing it. In theory the effective temperature is unique and independent of the chosen observable. This holds at the mean-field level, and Professor Ludovic Berthier's paper proves that it also holds, at least, in the Kob-Anderson supercooled liquid model^[5].

References

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1. Most likely caused by staying up late, which inevitably brings to mind a certain protagonist from a recent anime whose mental state is very similar ↩︎

2. Although I think this is entirely normal ↩︎

3. https://doi.org/10.1126/sciadv.abg6766 ↩︎

4. Among the old papers I have seen and still remember ↩︎

5. https://doi.org/10.1103/PhysRevLett.89.095702 ↩︎