Karakters - Netflix

Type: Documentary

Languages: Dutch

Status: Ended

Runtime: None minutes

Premier: 2013-11-11

Karakters - Brownian motion - Netflix

Brownian motion or pedesis (from Ancient Greek: πήδησις /pέːdεːsis/ “leaping”) is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the fast-moving molecules in the fluid. This pattern of motion typically alternates random fluctuations in a particle's position inside a fluid sub-domain with a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such fluid there exists no preferential direction of flow as in transport phenomena. More specifically the fluid's overall linear and angular momenta remain null over time. It is important also to note that the kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations sum up to the caloric component of a fluid’s internal energy. This motion is named after the botanist Robert Brown. In 1827, while looking through a microscope at particles trapped in cavities inside pollen grains in water, he noted that the particles moved through the water; but he was not able to determine the mechanisms that caused this motion. Atoms and molecules had long been theorized as the constituents of matter, and Albert Einstein published a paper in 1905 that explained in precise detail how the motion that Brown had observed was a result of the pollen being moved by individual water molecules, making one of his first big contributions to science. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist, and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 “for his work on the discontinuous structure of matter”. The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. In consequence only probabilistic models applied to molecular populations can be employed to describe it. Two such models of the statistical mechanics, due to Einstein and Smoluchowski are presented below. Another, pure probabilistic class of models is the class of the stochastic process models. There exist both simpler and more complicated stochastic processes which in extreme (“taken to the limit”) may describe the Brownian Motion (see random walk and Donsker's theorem).

Karakters - Einstein's theory - Netflix

x                                  2                                            ¯                                      2              t                                      =        D        =        μ                  k                      B                          T        =                                            μ              R              T                        N                          =                                            R              T                                      6              π              η              r              N                                      .              {\displaystyle {\frac {\overline {x^{2}}}{2t}}=D=\mu k_{B}T={\frac {\mu RT}{N}}={\frac {RT}{6\pi \eta rN}}.}  

ρ        (        x        ,        t        )        =                              N                          4              π              D              t                                                e                      −                                                            x                                      2                                                                    4                  D                  t                                                                    .              {\displaystyle \rho (x,t)={\frac {N}{\sqrt {4\pi Dt}}}e^{-{\frac {x^{2}}{4Dt}}}.}  

+                                                                                                    ∂                                                  2                                                                    ρ                                                              ∂                                              x                                                  2                                                                                                                    ⋅                                  ∫                                      −                    ∞                                                        +                    ∞                                                                                                              Δ                                              2                                                              2                                                  ⋅                φ                (                Δ                )                                                  d                                Δ                +                ⋯                                                                    =                

The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval. Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 1014 collisions per second. Thus Einstein was led to consider the collective motion of Brownian particles. He regarded the increment of particle positions in unrestricted one dimensional (x) domain as a random variable (

Karakters - References - Netflix