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## Exploiting finite-size effects in simulations

#### CECAM-FR-MOSER. Université Pierre et Marie Curie (UPMC), Paris

#### Organisers

Finite size effects are ubiquitous in simulations. Sources of such effects are of different nature: finite number of particles in a finite system, finite simulation box when using periodic boundary conditions, finite basis set in quantum chemistry, finite time step or grid spacing in algorithms, not even mentioning incomplete sampling of phase space due to the finite computational time.

Practical consequences of such finiteness are numerous, with dramatic implications on the computed properties and the (lack of) possibility to compare simulation results with experiments, which correspond in principle to an "infinite limit": thermodynamic limit (N→∞), hydrodynamic limit (k→0,ω →0), continuous limit (Δt→0, Δx→0), etc.

These effects are usually considered a nuisance and efficient strategy are developed to correct for these artefacts. However, they also reflect the physical properties of the system of interest. For example, some finite-size effects are due to the long range of the interactions (e.g. electrostatics), some due to the long range of the perturbations (e.g. strain fields and hydrodynamic flows), some due the long range of correlations (e.g. critical phenomena - but possibly also non-equilibrium effects).

In fact, it is also well known that in some cases one can exploit the size-dependence. In thermodynamics, examples include the detection first-order transitions or computation of (interfacial) free energies. In hydrodynamics computing the diffusion coefficient as a function of the inverse size of the system allows extrapolating to infinite size or as well as measuring the viscosity from the slope (in particular in ab initio MD, where other methods are too slow to converge). In crystals, finite size effects may be related to the fact that not every choice for the number of particles is compatible with the choice of boundary conditions, or to the finite size of solid particles. In a recent Quantum Monte Carlo approach to quantum chemistry, which involves the creation/annihilation of walkers in the space of Slater determinants, it is believed that the evolution of the number of walkers until convergence also reflects some intrinsic properties of the system under consideration.

This suggests that a new perspective on some finite-size effects (some are more "promising" than others) may provide a wealth of information on the physical properties, in a variety of contexts. In particular, organizing a discussion with open-minded experts in various aspects of simulation will be very beneficial for a broad range of problems. This includes revisiting some classics (which are nowadays to a large extent overlooked), such as the choice of boundary effects "at infinity" for the computation of electrostatic interactions or the dependence of computed properties and their fluctuations in the number of particles (and the thermodynamic ensemble), as well as exporting these ideas into new fields, such as rare events, glasses, or evolution, genetics and biochemical networks.

Finite-size effects in simulations are usually viewed as an inconvenience. However, often the magnitude of finite-size effects can provide unique physical information. The aim of the proposed workshop is to review the `positive power’ of finite-size effects and to discuss techniques to measure them.

Our discussion will not focus on how to eliminate finite-size effects. As the range of relevant computational problems with `interesting’ finite size effects is vast, we will focus on a set of representative problems.

Specifically, we have identified the following themes as key topics to be addressed during the workshop, which will serve to organize the presentations and discussions:

Finite-size effects and

• Interfacial phenomena

• Transport phenomena

• Quantum Monte Carlo simulations

• Thermodynamic ensemble, commensurability issues, finite particle size...

• Evolution, genetics, biochemical networks...

• "Infinite-size effects", i.e. boundary effects at infinity

• Fluctuation relations

• Algorithms, discretization...

We will not consider finite-size effects related to critical phenomena, as this is a subject that has been studied extensively in its own right.

## References

**France**

Benjamin Rotenberg (CNRS and Sorbonne Université) - Organiser

**United Kingdom**

Daan Frenkel (U Cambridge) - Organiser