We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If $p_3(n)$ denotes the number of solid partitions of an integer $n$, we show that $lim_{nrightarrowinfty} n^{-3/4} log p_3(n)sim 1.822pm 0.001$. Th
is shows clear deviation from the value $1.7898$, attained by MacMahon numbers $m_3(n)$, that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in $log p_3(n)$. In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious sub-leading terms. The period of the oscillatory term is proportional to $n^{1/4}$, the natural scale in the problem. This new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function.
Up-down asymmetric inclusions impose a local, spontaneous curvature to an elastic membrane. When several of them are inserted in a same membrane, they feel effective forces mediated by the membrane, both of elastic and entropic nature. Following an a
pproach initiated by Dommersnes and Fournier in the vanishing tension case [Eur. Phys. J. B 12, 9 (1999)], and also using a pseudo-analytical micellization theory, we derive the statistical mechanics of asymmetric inclusion assemblies when they are also subject to an additional short-range, attractive interaction. Our main conclusion is that generically, when the membrane is under tension, these inclusions live in small clusters at equilibrium, leading to local membrane invaginations. We also propose a novel curvature-induced demixing mechanism: when inclusions imposing local curvatures of opposite sign coexist, they tend to demix in distinct clusters under realistic conditions. This work has potential implications in the context of the thermodynamics of proteins embedded in biological lipid bilayers.
We propose a unifying, analytical theory accounting for the self-organization of colloidal systems in nano- or micro-cluster phases. We predict the distribution of cluter sizes with respect to interaction parameters and colloid concentration. In part
icular, we anticipate a proportionality regime where the mean cluster size grows proportionally to the concentration, as observed in several experiments. We emphasize the interest of a predictive theory in soft matter, nano-technologies and biophysics.
