No free lunch for effective potentials: general comment for Faraday FD144 arXiv:1001.1097v2 [cond-mat.soft] 7 Jan 2010 Ard A. Louis 1 $^{1}$Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, United Kingdom (Dated: November 17, 2018) I briefly review the problem of representability — a single coarse-grained effective pair potential cannot simultaneously represent all the properties of an underlying more complex system – as well as a few other subtleties that can arise in interpreting coarse-grained potentials. PACS numbers: an effective potential. The difference with the bare-pair potential w $^{(2)}$( r ) can be written as: An important source of progress in computer simula- tion comes from better coarse-grained models of the un- derlying materials, that is descriptions that are simpler and more tractable, but nevertheless retain the funda- mental underlying physics that one is interested in in- vestigating [1–3]. To most of us it is intuitively obvious that these simplified descriptions throw some information away and that compromises are made. After all, there is no such thing as a free lunch. But how much are we paying, and what can we get away with? Another popular way of deriving an effective poten- tial is by fitting to a thermodynamic observable like the internal energy. One first calculates the full internal en- ergy U ( N, V, T ) for the system governed by the Hamil- tonian (1), and then (in this case) derives a potential v eff U ( r ) that reproduces the same internal energy by the simpler two-body formula: To illustrate these questions, consider the example of a one-component reference fluid interacting with a three- body Hamiltonian of the form: I’ll call this the energy route . The difference with the bare-pair potential w $^{(2)}$( r ) can be written as: It is not hard to show that both v$_{g}$ ( r ) and v$_{U}$ ( r ) depend on the state point at which they are derived. Thus if they are used at a different state-point, one would expect transferability problems , i.e. you’d need to re-derive them for your new state-point. where r$_{i}$ denotes the position of particle i and r$_{ij}$ = r$_{i}$ − r$_{j}$ and r$_{ij}$ = | r$_{i}$ − r$_{j}$ | . K is the kinetic energy opera- tor, w $^{(2)}$( r ) is an isotropic pairwise additive potential, and w $^{(3)}$( r$_{ij}$ , r$_{jk}$ , r$_{ki}$ ) is a triplet or three-body potential. Three-body potentials are expensive and cumbersome to simulate, and so one might want to coarse-grain them to a simpler isotropic representation. What is perhaps more worrying is that one can also show that at a given state-point, v$_{g}$ ( r ) and v$_{U}$ ( r ) can- not be the same. As first demonstrated almost 40 years ago [8], to lowest order in ρ and w $^{(3)}$, the ratio between the two corrections is: In other words even in the low density weak potential limit, the corrections due to the three-body forces that you are coarse-graining out differ by a factor of three! What happens for stronger interactions or higher densi- ties needs to be investigated. Since v$_{g}$ ( r ) is unique, it is therefore impossible to represent all the properties of a system governed by the Hamiltonian of Eq. (1) by a sin- gle pair potential. There is no free lunch. I believe such There are several ways you could do this. One popular method is to fit a pair-potential such that it reproduces a structural quantity like the radial distribution function g ( r ) generated by the Hamiltonian of Eq. 1. Hender- son [4] first showed that “the pair potential v ( r ) which gives rise to a radial distribution function g ( r ) is unique up to a constant.”. A more rigourous mathematical dis- cussion is provided by Chayes, Chayes and Lieb [5]. An extended proof for orientational correlations can be found in a book by Gray and Gubbins [6] and we did this for multi-site potentials in ref. [7]. Therefore, for a given state-point, the g ( r ) generated by a Hamiltonian like that of Eq. (1) can be reproduced by a unique effective pair potential v$_{g}$ ( r ). (The existence of v$_{g}$ is more subtle, but holds under fairly general conditions, see [5]). I’ll call ap- proaches that derive v$_{g}$ ( r ) the structural route to deriving δv$_{U}$ ( r ) δv$_{g}$ ( r ) = 1 3 + O ( ( w $^{(3)}$)$^{2}$; ρ 2 ) . (5) B. representability problems A. coarse-graining: structure and energy routes U ( N, V, T ) = 1 2 ρ 2 ∫ d r$_{1}$ d r$_{2}$ g ( r$_{12}$ ) v eff U ( r$_{12}$ ) . (3) δv$_{U}$ ( r ) = w $^{(2)}$( r ) − v$_{U}$ ( r ) . (4) H = K + ∑ i<j w $^{(2)}$( r$_{ij}$ ) + ∑ i<j<k w $^{(3)}$( r$_{ij}$ , r$_{jk}$ , r$_{ki}$ ) , (1) δv$_{g}$ ( r ) = w $^{(2)}$( r ) − v$_{g}$ ( r ) . (2) I. INTRODUCTION