Related Experiment Video
Updated: Jun 22, 2026
04:13
Using a Real-Time Locating System to Measure Walking Activity Associated with Wandering Behaviors Among Institutionalized Older Adults
Published on: February 8, 2019
Localization of the maximal entropy random walk
1Marian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Kraków, Poland. zdzislaw.burda@uj.edu.pl
Physical Review Letters
|June 13, 2009
Summary
We introduce maximal entropy random walks, a new process that differs from generic random walks on irregular lattices. This novel random walk method shows particle localization in defect-free regions of diluted lattices.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
- Stochastic Processes
Background:
- Random walks are fundamental models for diffusion and transport.
- Entropy maximization is a key principle in statistical physics.
- Lattice irregularities can significantly alter random walk behavior.
Purpose of the Study:
- To define and investigate a novel class of random walk processes: maximal entropy random walks.
- To explore the behavior of these walks on irregular lattices, specifically those with weak dilution.
- To understand the underlying physical mechanisms driving observed phenomena.
Main Methods:
- Formulation of the maximal entropy random walk model.
- Analysis of the walk's behavior on a weakly diluted lattice.
- Application of Lifshitz state theory to explain localization.
Main Results:
- Maximal entropy random walks are equivalent to generic random walks on regular lattices.
- On irregular lattices, these walks exhibit distinct behavior.
- A clear localization phenomenon was observed, with stationary probability concentrating in large, defect-free regions.
Conclusions:
- Maximal entropy random walks offer a new perspective on stochastic processes in complex environments.
- The observed localization is a classical phenomenon driven by lattice structure.
- Lifshitz states provide a theoretical framework for understanding this localization in disordered systems.
Related Concept Videos
01:10Entropy Change in Reversible Processes
In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
01:14The Entropy as a State Function
Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
02:39Entropy
Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
01:18Entropy
The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
01:21Entropy Changes Accompanying Specific Processes
Entropy, a measure of disorder in a system, changes during phase transitions like freezing or boiling. At the transition temperature Ttrs, where two phases are in equilibrium, the phase transition is a reversible process. The entropy change can be calculated from a substance's enthalpy of transition using the equation ΔStrs = ΔtrsH /Ttrs.When a perfect gas expands isothermally from one volume to another, entropy increases logarithmically with volume. Conversely, isothermal compression results...
01:23Absolute Entropies and the Third Law of Thermodynamics
Ludwig Edward Boltzmann developed a definition for entropy, which stated that absolute entropy is proportional to the natural logarithm of the number of possible combinations of particles. Entropy stands alone among state functions as the only one whose absolute values can be determined.Consider a gas sample confined to a container. As the container expands, the energy levels of gas molecules become more closely spaced. This increases the number of available energy states, thereby increasing...