Abstract
Blending general relativity with Brownian motion theory leads to an interesting prediction: the classical root-of-time behavior of the RMS displacement of a Brownian particle is affected by gravity. These corrections to the diffusion law are not just a theoretical curiosity: they provide an opportunity for a new kind of “metamaterials”, where diffusive transport—as opposed to ray propagation, as in transformation optics—can be tailored with suitably designed effective metrics. What is more, this effect force us to reconsider the formulation of the second law of (non-equilibrium) thermodynamics in gravitational analogues, as tracers diffusing in curved effective metrics can sometimes accumulate (instead of spreading), thereby decreasing their Gibbs entropy, without any force being applied to them.
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- 1.
We choose units where the diffusion coefficient is 1.
- 2.
The covariant derivative D a associated to h ab acts on a tensor field \(T^{a_{1}\cdots a_{n}}_{\quad\quad b_{1}\cdots b_{m}}\) according to
$$\begin{aligned} D_{c}T^{a_{1}\cdots a_{n}}_{\quad\quad b_{1}\cdots b_{m}}=h^{\ a_{1}}_{e_{1}} \cdots h_{b_{m}}^{\ d_{m}} h^{\ f}_{c} \nabla_{f} T^{d_{1}\cdots d_{n}}_{\quad\quad e_{1}\cdots e_{m}}. \end{aligned}$$ - 3.
If spacetime is not static, this makes the transition rates implicit functions of time.
- 4.
I recommend van Kampen’s note [18] for a discussion of the applicability of this approximation.
- 5.
It is interesting to note that this equation is the same as the one postulated by Eckart [6] for thermal diffusion in his attempt to formulate a general-relativistic theory of dissipative hydrodynamics.
- 6.
That is not to say that the total probability is not conserved; we saw that it is.
- 7.
The original article [14] contains an error in this formula, pointed out by James Bonifacio at the University of Canterbury (New Zealand). I thank him for that.
- 8.
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Acknowledgements
The title of this chapter is inspired by the article [3], in which P. Castro-Villarreal computes the corrections of the RMS displacement of a Brownian particle in a curved space (as opposed to a curved spacetime, as in this chapter).
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Smerlak, M. (2013). Einstein2: Brownian Motion Meets General Relativity. In: Faccio, D., Belgiorno, F., Cacciatori, S., Gorini, V., Liberati, S., Moschella, U. (eds) Analogue Gravity Phenomenology. Lecture Notes in Physics, vol 870. Springer, Cham. https://doi.org/10.1007/978-3-319-00266-8_15
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