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Defining equation SI units Dimension Number of atoms N = Number of atoms remaining at time t. N 0 = Initial number of atoms at time t = 0 N D = Number of atoms decayed at time t = + dimensionless dimensionless Decay rate, activity of a radioisotope: A = Bq = Hz = s −1 [T] −1: Decay constant: λ
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). [2] This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume.
Note the minus sign in the equation, the drag force points in the opposite direction to the relative velocity: drag opposes the motion. Stokes' law makes the following assumptions for the behavior of a particle in a fluid: Laminar flow; No inertial effects (zero Reynolds number) Spherical particles; Homogeneous (uniform in composition) material
where denotes the surface normal direction of the solid structure, and the imaginary particle density is calculated through the equation of state. Accordingly, the interaction forces f a F : p {\displaystyle \mathbf {f} _{a}^{F:p}} and f a F : v {\displaystyle \mathbf {f} _{a}^{F:v}} acting on a solid particle a {\displaystyle a} are given by
The original Langevin equation [1] [2] describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid, = + (). Here, v {\displaystyle \mathbf {v} } is the velocity of the particle, λ {\displaystyle \lambda } is its damping coefficient, and m {\displaystyle m} is its mass.
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
This is called Abel's integral equation and allows us to compute the total time required for a particle to fall along a given curve (for which / would be easy to calculate). But Abel's mechanical problem requires the converse – given T ( y 0 ) {\displaystyle T(y_{0})\,} , we wish to find f ( y ) = d ℓ / d y {\displaystyle f(y)={d\ell }/{dy ...
The h-calculus is the calculus of finite differences, which was studied by George Boole and others, and has proven useful in combinatorics and fluid mechanics. In a sense, q -calculus dates back to Leonhard Euler and Carl Gustav Jacobi , but has only recently begun to find usefulness in quantum mechanics , given its intimate connection with ...
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