**Non-equilibrium Steady States**

**Long-range** (and universal) **Casimir forces** arise from **long-range correlated** (quantum or thermal) **fluctuations.**

Non-equilibrium fluctuations of conserved quantitiescan be long-ranged.Fluctuating hydrodynamics predicts long-range correlated temperature/density fluctuations.

H. Wada, and S.-i. Sasa, Phys. Rev. E

67, 065302 (2003) (Fluctuating shear flow)T. R. Kirkpatrick, J. M. Ortiz de Zárate, and J. V. Sengers, Phys. Rev. Lett.

110, 235902 (2013)Is there a corresponding

universalCasimir force?

**Diffusive Pressure?** Consider *diffusion* between reservoirs at different densities; i.e. a steady state with a uniform current.

"Fluctuation-induced forces in Non-equilibrium (difusive) dynamics," Aminov, Kafri, & M.K., PRL

114, 230602 (2015)

ρ_{l}_{ }ρ_{r}

The presence of a current leads to correlated, position dependent, fluctuations in density.

Density fluctuations are different on the 2 sides of each plate, leading to a position-dependent pressure

For the Symmetric Simple Exclusion Process (SSEP) in two dimensions

We tested this prediction numerically:

More generally, for small density gradients:

The amplitude is non-universal, and dependent on dynamics.

The force can be attractive (SSEP) or repulsive.

The same (non-extensive) form is obtained in higher dimensions

For plates a different temperatures, Kirkpatrick et al argue for a fluctuation-induced force:

**Radiation Pressure **arises in a non-equilibrium * steady state* with different temperatures

"Surface Phonon Polaritons Mediated Energy Transfer between Nanoscale Gaps," S.Shen, A. Narayanaswamy, & G. Chen, Nano Lett.

9, 2909 (2009) Breaking the law, at the nanoscale (MIT news, July 29, 2009)

A generalized approach for computation of Casimir forces, as well as radiation and heat transfer.

"Nonequilibrium Electromagnetic Fluctuations: Heat Transfer and Interactions,"

M. Krüger, T. Emig, and M. Kardar, Phys. Rev. Lett.

106, 210404 (2011)

Rytov (1959): "Fluctuational QED"

Fluctuating currents in each object are related to its temperature by a

fluctuation-dissipationcondition:

The EM field due to

thermal fluctuationsofone objectis related to overall Green's function by:

The overall fluctuations with many objects at different temperatures is then given by:

From EM correlations follow the stress tensor and the Poynting vector, hence forces and radiation.

Non-equilibrium steady states: Classical long-range forces & unusual radiative behavior.