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Register a new userInteraction Effects on the Size Distribution in a Growth Model
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We study, both analytically and numerically, the interaction effects on the skewness of the size distribution of elements in a growth model. We incorporate two types of global interaction into the growth model, and develop analytic expressions for the first few moments from which the skewness of the size distribution is calculated. It is found that depending on the sign of coupling, interactions may suppress or enhance the size growth, which in turn leads to the decrease or increase of the skewness. The amount of change tends to increase with the coupling strength, rather irrespectively of the details of the model.
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A detailed analysis of the finite-size effects on the bulk critical behaviour of the $d$-dimensional mean spherical model confined to a film geometry with finite thickness $L$ is reported. Along the finite direction different kinds of boundary conditions are applied: periodic $(p)$, antiperiodic $(a)$ and free surfaces with Dirichlet $(D)$, Neumann $(N)$ and a combination of Neumann and Dirichlet $(ND)$ on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary $d$. It is found, for $2<d<4$, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for $(p)$ and $(a)$. For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At $d=3$, the critical amplitude of the singular part of the free energy (related to the so called Casimir amplitude) is estimated. We obtain $Delta^{(p)}=-2zeta(3)/(5pi)=-0.153051...$, $Delta^{(a)}=0.274543...$ and $Delta^{(ND)}=0.01922...$, implying a fluctuation--induced attraction between the surfaces for $(p)$ and repulsion in the other two cases. For $(D)$ and $(N)$ we find a logarithmic dependence on $L$.
Using Monte Carlo simulations, finite-size effects of interfacial properties in the rough phase of the Ising on a cubic lattice with $Ltimes Ltimes R$ sites are studied. In particular, magnetization profiles perpendicular to the flat interface of size L$times$R are studied, with $L$ being considerably larger than $R$, in the (pre)critical temperature range. The resulting $R$-dependences are compared with predictions of the standard capillary-wave theory, in the Gaussian approximation, and with a field theory based on effective string actions, for $L$=$infty$.
We use the optimal fluctuation method to evaluate the short-time probability distribution $mathcal{P}left(H,L,tright)$ of height at a single point, $H=hleft(x=0,tright)$, of the evolving Kardar-Parisi-Zhang (KPZ) interface $hleft(x,tright)$ on a ring of length $2L$. The process starts from a flat interface. At short times typical (small) height fluctuations are unaffected by the KPZ nonlinearity and belong to the Edwards-Wilkinson universality class. The nonlinearity, however, strongly affects the (asymmetric) tails of $mathcal{P}(H)$. At large $L/sqrt{t}$ the faster-decaying tail has a double structure: it is $L$-independent, $-lnmathcal{P}simleft|Hright|^{5/2}/t^{1/2}$, at intermediately large $|H|$, and $L$-dependent, $-lnmathcal{P}sim left|Hright|^{2}L/t$, at very large $|H|$. The transition between these two regimes is sharp and, in the large $L/sqrt{t}$ limit, behaves as a fractional-order phase transition. The transition point $H=H_{c}^{+}$ depends on $L/sqrt{t}$. At small $L/sqrt{t}$, the double structure of the faster tail disappears, and only the very large-$H$ tail, $-lnmathcal{P}sim left|Hright|^{2}L/t$, is observed. The slower-decaying tail does not show any $L$-dependence at large $L/sqrt{t}$, where it coincides with the slower tail of the GOE Tracy-Widom distribution. At small $L/sqrt{t}$ this tail also has a double structure. The transition between the two regimes occurs at a value of height $H=H_{c}^{-}$ which depends on $L/sqrt{t}$. At $L/sqrt{t} to 0$ the transition behaves as a mean-field-like second-order phase transition. At $|H|<|H_c^{-}|$ the slower tail behaves as $-lnmathcal{P}sim left|Hright|^{2}L/t$, whereas at $|H|>|H_c^{-}|$ it coincides with the slower tail of the GOE Tracy-Widom distribution.
We present a renormalization approach to solve the Sznajd opinion formation model on complex networks. For the case of two opinions, we present an expression of the probability of reaching consensus for a given opinion as a function of the initial fraction of agents with that opinion. The calculations reproduce the sharp transition of the model on a fixed network, as well as the recently observed smooth function for the model when simulated on a growing complex networks.
Energy eigenvalues and order parameters are calculated by exact diagonalization for the transverse Ising model on square lattices of up to 6x6 sites. Finite-size scaling is used to estimate the critical parameters of the model, confirming universality with the three-dimensional classical Ising model. Critical amplitudes are also estimated for both the energy gap and the ground-state energy.
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