Periodic boundary conditions: torus or infinite images?

Question

I have a "philosophical" question regarding the use of periodic boundary conditions (PBD) in modeling and simulating systems of particles.

Let us consider a system of $N$ classical particles whose dynamics is described by Hamilton equations with the following Hamiltonian: $$ H(\boldsymbol{x}_1,...,\boldsymbol{x}_N,\boldsymbol{p}_1,...,\boldsymbol{p}_N) = \sum_{i=1}^N\frac{\boldsymbol{p}_i}{2m_i} + U(\boldsymbol{x}_1,...,\boldsymbol{x}_N) \; , $$ where $U$ is a generic interaction potential.

Usually, in methods like Molecular Dynamics (MD) the motion of particles is obtained via numerical integration of said Hamilton equations in a box with PBC using the minimum-image convention (MIC), that is defining the distance between two particles as the minimum distance between all the periodic images of said particles.

PBC + MIC have quite some important implications, as they affect the metric of the space in which the particles move. On a 'practical' standpoint, it is usually said that due to PCB + MIC it is not possible to cut-off long-range interactions (e.g Coulomb electrostatics, Newtonian gravity, etc...) due to the fact that the sum of said interactions over infinite periodic images is only conditionally convergent. This is why Ewald summation is needed to compute electrostatics in MD, for example.

Now, my question is: are there really infinite images? PBC can be either interpreted as "the systems moves on a torus" or "the system is replicated infinitely in a Cartesian space". Does using a special treatment of long-range interactions that assumes infinite periodic images single-out the latter interpretation and discard the former?

If the system moves on a torus, there's no need to compute the interactions over infinite images since all the images are the same image and there are truly only $N$ particles, so one can just directly sum over them. Or maybe the way I am approaching this question is completely wrong.

EDIT I: I thought I could also mention the reason why I am asking this question. In the famous book by Frenkel and Smith on molecular simulations [1] PBC are introduced to address the following problem:

In order to simulate bulk phases it is essential to choose boundary conditions that mimic the presence of an infinite bulk surrounding the N-particle model system

To me, an infinite bulk and a torus are conceptually very different! On a more practical note, I think the infinite bulk interpretation of PBC is suitable for material science, while the torus one is more suitable for biophysics.

[1] Daan Frenkel and Berend Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press 2002

EDIT II: After reading the answers I got so far, I think that what I should really be asking myself is why "long-range interactions wrap more around the torus" (quoting @Quillo). But that is probably a separate question.

Answer 1

PBC means that the system moves on a torus. This means that the particle positions $x$ are thought of as being defined up to a lattice $\Lambda$, i.e. living in $\mathbb R^D/\Lambda$. The issue is that your interaction term $U$ is therefore multivalued in this context, or in other words it is not $\Lambda$ periodic. You therefore need to periodise it to $U_\Lambda$, i.e. you want it to satisfy for all $\lambda_1,...,\lambda_N\in\Lambda$ and all $x_1,...,x_N\in\mathbb R^D$: $$ U_\Lambda(x_1,...,x_N) = U_\Lambda(x_1+\lambda_1,...,x_N+\lambda_N) $$ As a consistency condition, as the spatial periodicity goes to infinity, you should recover the original potential, so: $$ U_{L\Lambda} \xrightarrow{L\to\infty} U $$

There are many ways to do this. One way is to use your infinite images: $$ U_\Lambda(x_1,...,x_N) = \sum_{\lambda\in\Lambda^N}U(x_1+\lambda_1,...,x_N+\lambda_N) $$ It's a nice method if $U$ decays fast enough. In Fourier space, it is equivalent to sampling the Fourier transform on the dual lattice $\Lambda^*$. It also preserves smoothness etc. However when $U$ does not decay fast enough, you need to define the summation carefully.

The MIC is a separate method where you use the same periodising technique, but first you set $U$ to zero outside the Voronoi tile of the origin for $\Lambda^N$. In Fourier space, it is equivalent to sampling the Fourier transform on the dual lattice $\Lambda^*$ after having smoothed it out by convolving with a sinc filter. It's exactly like in sampling where you first filter the signal to avoid aliasing (except here you just use the brick wall filter). This has the consequence of typically creating "kinks" at the boundary of the Vornoi tiles but is easier to compute and does not require fancy resummation.

When the potential is long range, both methods will give rather different results. Depending on what you are interested in, one method may be more appropriate than another. An extreme example of this would be the 1D Coulomb potential for the interaction of two particles. In the first method, this will give periodic inverted parabolas, while in the second method, it will give a triangle wave. Notice that as expected, both agree at the minima, but the general structure is rather different.

The two methods are different, but since they both obey the same consistency condition in the infinite spatial period limit, they will give similar results in this limit. Since this is the limit you are interested in, because you usually want to model particles in space, for most practical purposes they coincide.

Hope this helps.