The Langevin Equation

Coffey, W. T.; Kalmykov, Yuri P.; Waldron, J T

doi:10.1142/9789812831965

Cited by 99 publications

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“…We start by considering an ensemble of spherical singledomain magnetic nanoparticles subjected to a time-varying magnetic field. There are two fundamental mechanisms governing the response of the nanoparticle magnetic dipole moment: (1) the entire particle rotates with respect to the fluid (Brownian or Debye relaxation), 16,18 while the magnetic dipole moment remains fixed with respect to the crystalline lattice; or (2) the magnetic dipole moment within the particle rotates with respect the crystalline lattice (Néel or internal relaxation). 16,19 Both of these mechanisms are responsible for the alignment of the magnetic dipole moment in the direction of the applied magnetic field.…”

Section: Methodsmentioning

confidence: 99%

“…There are two fundamental mechanisms governing the response of the nanoparticle magnetic dipole moment: (1) the entire particle rotates with respect to the fluid (Brownian or Debye relaxation), 16,18 while the magnetic dipole moment remains fixed with respect to the crystalline lattice; or (2) the magnetic dipole moment within the particle rotates with respect the crystalline lattice (Néel or internal relaxation). 16,19 Both of these mechanisms are responsible for the alignment of the magnetic dipole moment in the direction of the applied magnetic field. However, the alignment of the magnetic dipole moment with the external magnetic field is not complete due to the thermal interactions between the magnetic dipole moment and the surrounding environment, which contributes to a randomization of the dipole-moment orientation.…”

Section: Methodsmentioning

confidence: 99%

“…In the adiabatic limit of slowly varying magnetic fields, the behavior of magnetic nanoparticles in dilute ferrofluids can be understood via the Langevin formula. 16 That is, if the relaxation time is much less than the time scale of the timevarying magnetic field, the system can be treated as being in equilibrium at each instant of time. However, for longer relaxation times the system does not have time to reach an equilibrium state as the magnetic field changes and the effects of relaxation must be taken into account.…”

Section: Introductionmentioning

confidence: 99%

“…The relaxation time is governed by the fluid viscosity, η, the absolute temperature, T, and the hydrodynamic volume of the particle, V h . In terms of these quantities, the characteristic zero-field Brownian relaxation time is given by 16,18…”

Section: Introductionmentioning

confidence: 99%

“…In this case the relaxation time is governed by the saturation magnetization, M s , the temperature, T, the core volume, V c , the anisotropy constant, K, the damping constant, α , and the electron gyromagnetic ratio, γ . In terms of these quantities, the characteristic zero-field Néel relaxation time is given by 16,19…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Dependence of Brownian and Néel relaxation times on magnetic field strength

2013

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A simple treatment of Néel relaxation using the common zero-field relaxation time overestimates the relaxation time of the magnetization in situations relevant for MPI and MPS. For sinusoidally driven (or ramped) systems, whether or not a particular relaxation mechanism dominates or is even relevant depends on the magnetic field strength, the frequency (or ramp time), and the phase of the magnetization relative to the applied magnetic field.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dependence of Brownian and Néel relaxation times on magnetic field strength

2013

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Statistical mechanics of holonomic systems as a Brownian motion on smooth manifolds

2015

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The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein-Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology.

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A fractional motion diffusion model for a twice‐refocused spin‐echo pulse sequence

Karaman

Zhou

2018

NMR in Biomedicine

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The purpose of this study was to develop an analytical expression for a fractional motion (FM) diffusion model to characterize diffusion-induced signal attenuation in a twice-refocused spin-echo (TRSE) sequence that is resilient to eddy currents, and to demonstrate its applicability to human brain imaging in vivo. Based on the FM theory, which provides a unified statistical description for Langevin motions, the diffusion-weighted (DW) MR signal was measured with a TRSE sequence that balances the concomitant gradients. The analytical expression was fitted to a set of DW images acquired with 14 b-values (0-4000 s/mm ) from a total of 10 healthy human subjects at 3 T, yielding three FM parameter maps based on anomalous diffusion coefficient D , diffusion increment variance φ, and diffusion correlation ψ, respectively. These parameters were used to characterize different brain regions in gray matter (GM), white matter (WM), and cerebrospinal fluid. The analytical expression for the TRSE-based FM model accurately described diffusion signal attenuation in healthy brain tissues at high b-values. TRSE's robustness against eddy currents was illustrated by comparing results from an expression for a conventional Stejskal-Tanner sequence. The TRSE-based FM model also produced consistent GM-WM contrast (p < 0.01) across all brain regions studied, whereas the consistency was not observed with the Stejskal-Tanner-based FM model. This new analytical expression is expected to enable further investigations to probe tissue structures by exploiting anomalous diffusion properties without being hindered by eddy-current perturbations at high b-values.

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The Langevin Equation

Cited by 99 publications

References 0 publications

Dependence of Brownian and Néel relaxation times on magnetic field strength

Dependence of Brownian and Néel relaxation times on magnetic field strength

Statistical mechanics of holonomic systems as a Brownian motion on smooth manifolds

A fractional motion diffusion model for a twice‐refocused spin‐echo pulse sequence

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