Related Experiment Video
Updated: May 18, 2026
11:03
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Matrix-valued Boltzmann equation for the Hubbard chain.
Martin L R Fürst1, Christian B Mendl, Herbert Spohn
1Zentrum Mathematik, Boltzmannstrasse 3, Technische Universität München and Excellence Cluster Universe, Boltzmannstrasse 2, 85748 Garching bei München, Germany. mfuerst@ma.tum.de
Summary
We analyzed the Boltzmann transport equation for the Hubbard chain, finding it reaches stationary states quickly. This study characterizes these nonthermal states and their convergence rates.
Area of Science:
- Condensed matter physics
- Statistical mechanics
- Quantum transport
Background:
- The Hubbard chain is a fundamental model in condensed matter physics.
- Understanding transport properties is crucial for materials science.
- Integrable systems exhibit unique properties like nonthermal stationary states.
Purpose of the Study:
- To analytically and numerically study the Boltzmann transport equation for the Hubbard chain.
- To characterize all stationary solutions, including nonthermal ones.
- To investigate the convergence rate to stationarity.
Main Methods:
- Analytical solutions of the Boltzmann transport equation.
- Numerical simulations of the time-dependent Wigner function.
- Analysis of spin-dependent matrix-valued Wigner functions.
Main Results:
- The H theorem is shown to hold.
- Infinitely many conservation laws indicate the integrability of the nearest-neighbor chain.
- Nonthermal stationary states were identified and characterized.
- Exponentially fast convergence to stationarity was observed numerically.
Conclusions:
- The Hubbard chain exhibits complex transport behavior due to its integrability.
- Stationary states, including nonthermal ones, are reachable.
- Convergence to stationarity is rapid and depends on initial conditions.
Related Concept Videos
01:20Maxwell-Boltzmann Distribution: Problem Solving
Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
01:29Chain Reactions
Chain reactions involve highly reactive transient species, such as atoms or free radicals, as intermediates. These intermediates facilitate rapid reactions over an extended period. The process includes a series of steps: a reactive intermediate is consumed, reactants are converted to products, and the intermediate is regenerated. This cycle enables continuous repetition, amplifying the production of products with a small amount of intermediate. Chain reactions often utilize free radicals as...
01:17Radical Chain-Growth Polymerization: Chain Branching
The skeletal structure of polymers synthesized via radical polymerization is always branched. For example, the polymerization of ethylene by radical polymerization results in a low-density grade of polyethylene with a heavily branched skeletal structure. Here, the radical site abstracts hydrogen from the growing chain, and the radical site shifts from the end (a primary carbon center) to anywhere within the growing chain (a secondary carbon center). Consequently, the part of the chain from the...
00:59Bernoulli's Equation
In the middle of the nineteenth century, it was observed that two trains passing each other at a high relative speed get pulled towards each other. The same occurs when two cars pass each other at a high relative speed. The reason is that the fluid pressure drops in the region where the fluid speeds up. As the air between the trains or the cars increases in speed, its pressure reduces. The pressure on the outer parts of the vehicles is still the atmospheric pressure, while the resultant...
01:23Equilibrium Conditions for a Particle
When an object is in equilibrium, it is either at rest or moving with a constant velocity. There are two types of equilibrium: static and dynamic. Static equilibrium occurs when an object is at rest, while dynamic equilibrium occurs when an object is moving with a constant velocity. In both cases, there must be a balance of forces acting on the object.
To understand the concept of equilibrium, let us first consider the forces acting on an object. When different forces act on an object, they can...
To understand the concept of equilibrium, let us first consider the forces acting on an object. When different forces act on an object, they can...
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.