No Arabic abstract
We investigate how the range of parameters that specify the two-particle distribution function is restricted if we require that this function be obtained from the $n^{rm th}$ order distribution functions that are symmetric with respect to the permutation of any two particles. We consider the simple case when each variable in the distribution functions can take only two values. Results for all $n$ values are given, including the limit of $ntoinfty$. We use our results to obtain bounds on the allowed values of magnetization and magnetic susceptibility in an $n$ particle Fermi fluid.
The local number variance associated with a spherical sampling window of radius $R$ enables a classification of many-particle systems in $d$-dimensional Euclidean space according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To better characterize density fluctuations, we carry out an extensive study of higher-order moments, including the skewness $gamma_1(R)$, excess kurtosis $gamma_2(R)$ and the corresponding probability distribution function $P[N(R)]$ of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform models. Specifically, we derive explicit integral expressions for $gamma_1(R)$ and $gamma_2(R)$ involving up to three- and four-body correlation functions, respectively. We also derive rigorous bounds on $gamma_1(R)$, $gamma_2(R)$ and $P[N(R)]$. High-quality simulation data for these quantities are generated for each model. We also ascertain the proximity of $P[N(R)]$ to the normal distribution via a novel Gaussian distance metric $l_2(R)$. Among all models, the convergence to a central limit theorem (CLT) is generally fastest for the disordered hyperuniform processes. The convergence to a CLT is slower for standard nonhyperuniform models, and slowest for the antihyperuniform model studied here. We prove that one-dimensional hyperuniform systems of class I or any $d$-dimensional lattice cannot obey a CLT. Remarkably, we discovered that the gamma distribution provides a good approximation to $P[N(R)]$ for all models that obey a CLT, enabling us to estimate the large-$R$ scalings of $gamma_1(R)$, $gamma_2(R)$ and $l_2(R)$. For any $d$-dimensional model that decorrelates or correlates with $d$, we elucidate why $P[N(R)]$ increasingly moves toward or away from Gaussian-like behavior, respectively.
The asymptotic analytic expression for the two-time free energy distribution function in (1+1) random directed polymers is derived in the limit when the two times are close to each other
We study the probability distribution $P(X_N=X,N)$ of the total displacement $X_N$ of an $N$-step run and tumble particle on a line, in presence of a constant nonzero drive $E$. While the central limit theorem predicts a standard Gaussian form for $P(X,N)$ near its peak, we show that for large positive and negative $X$, the distribution exhibits anomalous large deviation forms. For large positive $X$, the associated rate function is nonanalytic at a critical value of the scaled distance from the peak where its first derivative is discontinuous. This signals a first-order dynamical phase transition from a homogeneous `fluid phase to a `condensed phase that is dominated by a single large run. A similar first-order transition occurs for negative large fluctuations as well. Numerical simulations are in excellent agreement with our analytical predictions.
We derive a hierarchy of equations which allow a general $n$-body distribution function to be measured by test-particle insertion of between $1$ and $n$ particles, and successfully apply it to measure the pair and three-body distribution functions in a simple fluid. The insertion-based methods overcome the drawbacks of the conventional distance-histogram approach, offering enhanced structural resolution and a more straightforward normalisation. They will be especially useful in characterising the structure of inhomogeneous fluids and investigating closure approximations in liquid state theory.
We consider a single run-and-tumble particle (RTP) moving in one dimension. We assume that the velocity of the particle is drawn independently at each tumbling from a zero-mean Gaussian distribution and that the run times are exponentially distributed. We investigate the probability distribution $P(X,N)$ of the position $X$ of the particle after $N$ runs, with $Ngg 1$. We show that in the regime $ X sim N^{3/4}$ the distribution $P(X,N)$ has a large deviation form with a rate function characterized by a discontinuous derivative at the critical value $X=X_c>0$. The same is true for $X=-X_c$ due to the symmetry of $P(X,N)$. We show that this singularity corresponds to a first-order condensation transition: for $X>X_c$ a single large jump dominates the RTP trajectory. We consider the participation ratio of the single-run displacements as the order parameter of the system, showing that this quantity is discontinuous at $X=X_c$. Our results are supported by numerical simulations performed with a constrained Markov chain Monte Carlo algorithm.