![]() | This article provides insufficient context for those unfamiliar with the subject.(June 2021) |
In dynamics, probability, physics, chemistry and related fields, a heterogeneous random walk in one dimension is a random walk in a one dimensional interval with jumping rules that depend on the location of the random walker in the interval.
For example: say that the time is discrete and also the interval. Namely, the random walker jumps every time step either left or right. A possible heterogeneous random walk draws in each time step a random number that determines the local jumping probabilities and then a random number that determines the actual jump direction. Specifically, say that the interval has 9 sites (labeled 1 through 9), and the sites (also termed states) are connected with each other linearly (where the edges sites are connected their adjacent sites and together). In each time step, the jump probabilities (from the actual site) are determined when flipping a coin; for head we set: probability jumping left =1/3, where for tail we set: probability jumping left = 0.55. Then, a random number is drawn from a uniform distribution: when the random number is smaller than probability jumping left, the jump is for the left, otherwise, the jump is for the right. Usually, in such a system, we are interested in the probability of staying in each of the various sites after t jumps, and in the limit of this probability when t is very large, .
Generally, the time in such processes can also vary in a continuous way, and the interval is also either discrete or continuous. Moreover, the interval is either finite or without bounds. In a discrete system, the connections are among adjacent states. The basic dynamics are either Markovian, semi-Markovian, or even not Markovian depending on the model. In discrete systems, heterogeneous random walks in 1d have jump probabilities that depend on the location in the system, and/or different jumping time (JT) probability density functions (PDFs) that depend on the location in the system.[ citation needed ] ,,,,,,,,,,,,,,,,, General solutions for heterogeneous random walks in 1d obey equations ( 1 )-( 5 ), presented in what follows.
Random walks [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] can be used to describe processes in biology, [12] [ failed verification ] chemistry, [13] and physics, [14] [15] including chemical kinetics [13] and polymer dynamics. [14] [15] In ndividual molecules, random walks appear when studying individual molecules, [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] individual channels, [26] [27] individual biomolecules, [28] individual enzymes, [18] [20] [21] [22] [29] [30] [31] [32] [33] and quantum dots. [34] [35] [36] Importantly, PDFs and special correlation functions[ clarification needed ] can be easily calculated from single molecule measurements but not from ensemble measurements. This unique information can be used for discriminating between distinct random walk models that share some properties[ which? ], [18] [19] [20] [21] [22] [23] [24] [25] [26] and this demands a detailed theoretical analysis of random walk models. In this context, utilizing the information content in single molecule data is a matter of ongoing research.[ weasel words ]
The actual random walk obeys a stochastic equation of motion, but its probability density function (PDF) obeys a deterministic equation. PDFs of random walks can be formulated in terms of the (discrete in space) master equation [1] [12] [13] and the generalized master equation [3] or the (continuous in space and time) Fokker Planck equation [37] and its generalizations. [10] Continuous time random walks, [1] renewal theory, [38] and the path representation [3] [6] [8] [9] are also useful formulations of random walks. The network of relationships between the various descriptions provides a powerful tool in the analysis of random walks. Arbitrarily heterogeneous environments make the analysis difficult, especially in high dimensions.[ weasel words ]
Known important results in simple systems include:
The solution for the Green's function for a semi-Markovian random walk in an arbitrarily heterogeneous environment in 1D was recently given using the path representation. [6] [8] [9] (The function is the PDF for occupying state i at time t given that the process started at state j exactly at time 0.) A semi-Markovian random walk in 1D is defined as follows: a random walk whose dynamics are described by the (possibly) state- and direction-dependent JT-PDFs, , for transitions between states i and i ± 1, that generates stochastic trajectories of uncorrelated waiting times that are not-exponential distributed. obeys the normalization conditions (see fig. 1)
The dynamics can also include state- and direction-dependent irreversible trapping JT-PDFs, , with I=i+L. The environment is heterogeneous when depends on i. The above process is also a continuous time random walk and has an equivalent generalized master equation representation for the Green's function.. [3] [6] [8] [9]
In a completely heterogeneous semi-Markovian random walk in a discrete system of L (> 1) states, the Green's function was found in Laplace space (the Laplace transform of a function is defined with, ). Here, the system is defined through the jumping time (JT) PDFs: connecting state i with state j (the jump is from state i). The solution is based on the path representation of the Green's function, calculated when including all the path probability density functions of all lengths:
| (1) |
Here,
and
Also, in Eq. ( 1 ),
| (2) |
and
| (3) |
with
| (4) |
and
| (5) |
For L = 1, . In this paper, the symbol [L/2], as appearing in the upper bound of the sum in eq. ( 5 ) is the floor operation (round towards zero). Finally, the factor in eq. ( 1 ) has the same form as in in eqs. ( 3 )-( 5 ), yet it is calculated on a lattice . Lattice is constructed from the original lattice by taking out from it the states i and j and the states between them, and then connecting the obtained two fragments. For cases in which a fragment is a single state, this fragment is excluded; namely, lattice is the longer fragment. When each fragment is a single state, .
Equations ( 1 )-( 5 ) hold for any 1D semi-Markovian random walk in a L-state chain, and form the most general solution in an explicit form for random walks in 1d.
Clearly, in Eqs. ( 1 )-( 5 ) solves the corresponding continuous time random walk problem and the equivalent generalized master equation. Equations ( 1 )-( 5 ) enable analyzing semi-Markovian random walks in 1D chains from a wide variety of aspects. Inversion to time domain gives the Green’s function, but also moments and correlation functions can be calculated from Eqs. ( 1 )-( 5 ), and then inverted into time domain (for relevant quantities). The closed-form also manifests its utility when numerical inversion of the generalized master equation is unstable. Moreover, using in simple analytical manipulations gives, [6] [8] [9] (i) the first passage time PDF, (ii)–(iii) the Green’s functions for a random walk with a special WT-PDF for the first event and for a random walk in a circular L-state 1D chain, and (iv) joint PDFs in space and time with many arguments.
Still, the formalism used in this article is the path representation of the Green's function , and this supplies further information on the process. The path representation follows:
| (6) |
The expression for in Eq. ( 6 ) follows,
| (7) |
is the PDF of reaching state i exactly at time t when starting at state j exactly at time 0. This is the path PDF in time that is built from all paths with transitions that connect states j with i. Two different path types contribute to : [8] [9] paths made of the same states appearing in different orders and different paths of the same length of transitions. Path PDFs for translation invariant chains are mono-peaked. Path PDF for translation invariant chains mostly contribute to the Green's function in the vicinity of its peak, but this behavior is believed to characterize heterogeneous chains as well.
We also note that the following relation holds, . Using this relation, we focus in what follows on solving .
Complementary information on the random walk with that supplied with the Green’s function is contained in path PDFs. This is evident, when constructing approximations for Green’s functions, in which path PDFs are the building blocks in the analysis. [8] [9] Also, analytical properties of the Green’s function are clarified only in path PDF analysis. Here, presented is the recursion relation for in the length n of path PDFs for any fixed value of L. The recursion relation is linear in path PDFs with the s in Eq. ( 5 ) serving as the n independent coefficients, and is of order [L / 2]:
| (8) |
The recursion relation is used for explaining the universal formula for the coefficients in Eq. ( 1 ). The solution of the recursion relation is obtained by applying a z transform:
| (9) |
Setting in Eq. ( 9 ) gives . The Taylor expansion of Eq. ( 9 ) gives . The result follows:
| (10) |
In Eq. ( 10 ) is one for , and otherwise,
| (11) |
where
| (12) |
The initial number follow:
| (13) |
and,
| (14) |
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov.
In quantum mechanics, a density matrix is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.
In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an "observation". It is the essence of a measurement in quantum mechanics which connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is the continuous evolution via the Schrödinger equation. Collapse is a black box for a thermodynamically irreversible interaction with a classical environment. Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions apparently reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation throughout this apparent collapse. More importantly, this is not enough to explain actual wave function collapse, as decoherence does not reduce it to a single eigenstate.
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.
In mathematics, a random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.
The Ising model, , named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states. The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields.
A Lévy flight is a random walk in which the step-lengths have a Lévy distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. Later researchers have extended the use of the term "Lévy flight" to also include cases where the random walk takes place on a discrete grid rather than on a continuous space.
The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics. It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a Poisson equation. In the latter case it is also referred to in the plural form.
In condensed matter physics, Anderson localization is the absence of diffusion of waves in a disordered medium. This phenomenon is named after the American physicist P. W. Anderson, who was the first to suggest that electron localization is possible in a lattice potential, provided that the degree of randomness (disorder) in the lattice is sufficiently large, as can be realized for example in a semiconductor with impurities or defects.
Attosecond physics, also known as attophysics, or more generally attosecond science, is a branch of physics that deals with light-matter interaction phenomena wherein attosecond photon pulses are used to unravel dynamical processes in matter with unprecedented time resolution.
In probability theory, the Gillespie algorithm generates a statistically correct trajectory of a stochastic equation system for which the reaction rates are known. It was created by Joseph L. Doob and others, presented by Dan Gillespie in 1976, and popularized in 1977 in a paper where he uses it to simulate chemical or biochemical systems of reactions efficiently and accurately using limited computational power. As computers have become faster, the algorithm has been used to simulate increasingly complex systems. The algorithm is particularly useful for simulating reactions within cells, where the number of reagents is low and keeping track of the position and behaviour of individual molecules is computationally feasible. Mathematically, it is a variant of a dynamic Monte Carlo method and similar to the kinetic Monte Carlo methods. It is used heavily in computational systems biology.
In computational physics, variational Monte Carlo (VMC) is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system.
The Ghirardi–Rimini–Weber theory (GRW) is a spontaneous collapse theory in quantum mechanics, proposed in 1986 by Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber.
Diffusion is the net movement of anything generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, like in spinodal decomposition.
The term file dynamics is the motion of many particles in a narrow channel.
The Koopman–von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively.
In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times. More generally it can be seen to be a special case of a Markov renewal process.
Maximal entropy random walk (MERW) is a popular type of biased random walk on a graph, in which transition probabilities are chosen accordingly to the principle of maximum entropy, which says that the probability distribution which best represents the current state of knowledge is the one with largest entropy. While standard random walk chooses for every vertex uniform probability distribution among its outgoing edges, locally maximizing entropy rate, MERW maximizes it globally by assuming uniform probability distribution among all paths in a given graph.