[...]... the learning process But uncertainties are always there Too often these uncertainties are ignored and their study delayed or omitted altogether An Introduction to Stochastic Processes in Physics revisits elementary and foundational problems in classical physics and reformulates them in the language of random variables Well-characterized random variables quantify uncertainty and tell us what can be known... Albert Einstein had, three years earlier than Langevin, quantified Brownian motion with different methods, but we adopt Langevin’s approach because it builds most directly on Newtonian dynamics and on concepts familiar from elementary physics Indeed, Langevin claimed his method was in nitely more simple” than Einstein’s In 1943 Subrahmanyan Chandrasekhar was able to solve a number of important dynamical... the top of the trajectory In this as well as in similar cases we distinguish between systematic error and random variation The former can, in principle, be understood and quantified and thereby controlled or eliminated Truly random sources of variation cannot be associated with determinate physical causes and are often too small to be directly observed Yet, unnoticeably small and unknown random in uences... Suppose the box contains 10 nominally 5-Ohm resistors, each with a 20% tolerance Calculate the mean, variance, and tolerance of the resistance of their series combination Is the tolerance of this combination less than the tolerance of the separate resistors? It should be 2.4 Density Fluctuations The molecular number density ρ = N /V of a gas contained in a small open region of volume V within a larger closed... before quantum mechanics was discovered in the 1920s Classical uncertainty preceded quantum uncertainty because, unlike the latter, the former is rooted in easily recognized human conditions We are too small and the universe too large and too interrelated for thoroughly deterministic thinking For whatever reason—fundamental physical indeterminism, human finitude, or both—there is much we don’t know And what... dimensionless “tolerance” or “precision” ti whose definition we take to be √ var{Ri } × 100% ti = mean{Ri } where i = 1 n Assume the resistances Ri are statistically independent random variables PROBLEMS 15 a Write expressions for the mean, variance, and tolerance of the total resistance R of a series combination of n identically defined resistors in terms of the mean{Ri } and tolerance ti of one resistor b Suppose... Harrington, Ed Staneck, and Don Quiring made valuable comments on various parts of the typescript Willis Overholt helped with the figures More general encouragement came from Reuben Hersh, Arnold Wedel, and Anthony Gythiel I am grateful for all of these friends An Introduction to Stochastic Processes in Physics 1 Random Variables 1.1 Random and Sure Variables A quantity that, under given conditions, can... propellant Even when you are careful to keep the tube at a constant elevation, to inject the same quantity of propellant, and to keep all conditions constant, the projectile lands at noticeably different places in several trials One can imagine a number of causes of this variation: different initial orientations of a not-exactly-spherical ball, slightly variable amounts of propellant, and breeziness... human eye cannot distinguish the effect of individual molecular impacts, it can observe the net motion caused by many impacts over a period of time 18 RANDOM STEPS 3.2 Brownian Motion Modeled Let’s model Brownian motion as a sum of independent random displacements Imagine the Brownian particle starts at the origin x = 0 and is free to move in either direction along the x-axis The net effect of many individual... know is tinged with uncertainty Baseballs and hydrogen atoms behave, to a greater or lesser degree, unpredictably Uncertainties attend their initial conditions and their dynamical evolution This also is true of every artificial device, natural system, and physics experiment Nevertheless, physics and engineering curriculums routinely invoke precise initial conditions and the existence of deterministic . alt="" An Introduction to Stochastic Processes in Physics