# A quantum diffusion law

## TL;DR

This paper derives a quantum diffusion law using the fluctuation-dissipation theorem, revealing multiple regimes of diffusion at low temperatures and confirming logarithmic spreading behavior as robust, with potential for experimental testing.

## Contribution

It introduces a physically natural response function satisfying key physical conditions and demonstrates the equivalence of Wightman positivity and passivity under FDT, advancing understanding of quantum diffusion.

## Key findings

- Logarithmic spreading in the quantum regime is robust.
- The response function R(t) satisfies Wightman positivity and passivity.
- The diffusion law can be tested with ultra cold atom experiments.

## Abstract

We analyse diffusion at low temperature by bringing the fluctuation-dissipation theorem (FDT) to bear on a physically natural, viscous response-function R(t). The resulting diffusion-law exhibits several distinct regimes of time and temperature, each with its own characteristic rate of spreading. As with earlier analyses, we find logarithmic spreading in the quantum regime, indicating that this behavior is robust. A consistent R(t) must satisfy the key physical requirements of Wightman positivity and passivity, and we prove that ours does so. We also prove in general that these two conditions are equivalent when the FDT holds. Given current technology, our diffusion law can be tested in a laboratory with ultra cold atoms.