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Related Concept Videos

Atomic Nuclei: Types of Nuclear Relaxation01:28

Atomic Nuclei: Types of Nuclear Relaxation

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Nuclear relaxation restores the equilibrium population imbalance and can occur via spin–lattice or spin–spin mechanisms, which are first-order exponential decay processes.
In spin–lattice or longitudinal relaxation, the excited spins exchange energy with the surrounding lattice as they return to the lower energy level. Among several mechanisms that contribute to spin–lattice relaxation, magnetic dipolar interactions are significant. Here, the excited nucleus transfers...
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Nonstandard Reaction Conditions
The interconnection between standard cell potentials and various thermodynamic parameters such as the standard free energy change ΔG° and equilibrium constant K has been previously explored. For example, a redox reaction involving zinc(II) and tin(II) ions at 1 M concentration with Eºcell = +0.291 V and ΔG° = −56.2 kJ is spontaneous.
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Atomic Nuclei: Nuclear Relaxation Processes01:23

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In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis,  the precessing magnetic moments are randomly oriented around the z-axis.
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Atomic Nuclei: Nuclear Spin State Population Distribution01:14

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Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N−) and spin +½ (N+) states.
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Maxwell's Thermodynamic Relations01:23

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Maxwell's thermodynamic relations are very useful in solving problems in thermodynamics. Each of Maxwell's relations relates a partial differential between quantities that can be hard to measure experimentally to a partial differential between quantities that can be easily measured. These relations are a set of equations derivable from the symmetry of the second derivatives and the thermodynamic potentials.
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Systems in mechanical equilibrium exert equal pressure on the separating wall. Similarly, systems in thermal equilibrium share a common thermodynamic property: temperature.Temperature is a measure of the average kinetic energy of particles within a system. More generally, it reflects the internal energy state of the system. The higher the temperature, the more energy a system has, given that other variables, such as volume and pressure, remain constant. However, temperature is not a form of...
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Author Spotlight: Exploring Intrinsically Disordered Protein Dynamics Through NMR Relaxation Experiments
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Tensor-network algorithm for nonequilibrium relaxation in the thermodynamic limit.

Yoshihito Hotta1

  • 1Department of Physics, University of Tokyo, Komaba, Meguro, Tokyo 153-8505, Japan.

Physical Review. E
|July 15, 2016
PubMed
Summary

We developed a new tensor-network algorithm to study how systems reach equilibrium. This method precisely simulates the time evolution of magnetization in large systems, aiding critical phenomena research.

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Area of Science:

  • Statistical Physics
  • Computational Physics
  • Condensed Matter Theory

Background:

  • Studying nonequilibrium stochastic dynamics in the thermodynamic limit is computationally challenging.
  • Traditional methods often require long simulations or approximations for large systems.

Purpose of the Study:

  • To introduce a novel tensor-network algorithm for simulating discrete-time stochastic dynamics.
  • To apply this algorithm to analyze nonequilibrium relaxation in Ising models.

Main Methods:

  • Mapping d-dimensional Markov processes to (d+1)-dimensional infinite tensor networks using higher-order singular-value decomposition.
  • Utilizing translational invariance for direct analysis in the thermodynamic limit.
  • Applying the nonequilibrium-relaxation method framework.

Main Results:

  • Successfully computed the nonequilibrium relaxation dynamics for 1D and 2D Ising models.
  • Estimated the dynamical critical exponent for the 2D Ising model as z=2.16(5).
  • Demonstrated precise computation of magnetization time evolution for large systems over short periods.

Conclusions:

  • The proposed tensor-network algorithm offers an efficient approach for studying critical phenomena.
  • This method complements the nonequilibrium-relaxation method by enabling precise simulations of large systems.
  • The algorithm provides a new computational tool for investigating complex dynamical systems.