A Rheological Analogue for Brownian Motion with Hydrodynamic Memory

Abstract

When the density of the fluid surrounding suspended Brownian particles is appreciable, in addition to the forces appearing in the traditional Ornstein and Uhlenbeck theory of Brownian motion, additional forces emerge as the displaced fluid in the vicinity of the randomly moving Brownian particle acts back on the particle giving rise to long-range force correlations which manifest as a ``long-time tail'' in the decay of the velocity autocorrelation function known as hydrodynamic memory. In this paper, after recognizing that for Brownian particles immersed in a Newtonian, viscous fluid, the hydrodynamic memory term in the generalized Langevin equation is essentially the 1/2 fractional derivative of the velocity of the Brownian particle, we present a rheological analogue for Brownian motion with hydrodynamic memory which consists of a linear dashpot of a fractional Scott-Blair element and an inerter. The synthesis of the proposed mechanical network that is suggested from the structure of the generalized Langevin equation simplifies appreciably the calculations of the mean-square displacement and its time-derivatives which can also be expressed in terms of the two-parameter Mittag--Leffler function.

arXiv commentarXiv admin note: text overlap with arXiv:2102.01786

References in corpus (13)

• Brownian motion at short time scales
• Anisotropic memory effects in confined colloidal diffusion
• Ageing underdamped scaled Brownian motion: ensemble and time averaged particle displacements, non-ergodicity, and the failure of the overdamping approximation
• Persistent correlation of constrained colloidal motion
• Nearly-logarithmic decay in the colloidal hard-sphere system
• Accounting for inertia effects to access the high-frequency microrheology of viscoelastic fluids
• Short-time inertial response of viscoelastic fluids measured with Brownian motion and with active probes
• Molecular dynamics of partially confined Lennard-Jones gases: Velocity autocorrelation function, mean squared displacement, and collective excitations
• Time-Response Functions of Mechanical Networks with Inerters and Causality
• The fractional derivative of the Dirac delta function and new results on the inverse Laplace transform of irrational functions
• Viscous-Viscoelastic Correspondence Principle for Brownian Motion
• Time-Response Functions of Fractional-Derivative Rheological Models
• Impulse Response Function for Brownian Motion

Cited by in corpus (5)

• Mechanical Analog for Cities
• Energy fluctuations of a Brownian particle freely moving in a liquid
• Fractional and scaled Brownian motion on the sphere: The effects of long-time correlations on navigation strategies
• Nonequilibrium dynamical structure of a dilute suspension of active particles in a viscoelastic fluid
• Brownian Particles and Matter Waves