Dimensional Analysis of Walking

Let s summarize what we have found for the pendulum. Dimensional analysis can simplify the analysis of a model. But dimensional analysis is limited eventually we must go to the lab. However, our experimental agenda is shortened by dimensional analysis. In addition, our analysis of the data will be simplified because we know how quantities are grouped. 

It is worth noting that our analysis of the pendulum was aided by some key experiments that preceded ours. Specifically, we benefited from experiments that determined physical constants. In 1604, Galileo Galilei sought a formula for the time a mass m falls a distance h  

Galileo discovered that tfaii was independent of mass and proposed a constant to relate the dependence of feii on h. The constant is g. Other key experiments have provided the universal gravitational constant, Planck s constant, Boltzmann s constant, and the speed of light. 

Let s apply dimensional analysis to another phenomenon - walking. Since all humans are constructed similarly, we should be able to predict a person s walking velocity based on physical parameters. Clearly this is a system that would be difficult to analyze at a fundamental level. 

We begin by forming a list of the possible parameters of walking. The first parameter is what we seek - velocity. We start a table similar to that of the pendulum analysis. 

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The dimensional analysis of walking demonstrates an important concept in dynamic similarity. The magnitude of a dimensionless group indicates the character of the phenomenon. That is, if you are told that a walker had a Froude number of 2.0,... 

The microscopic view of diffusion starts with the movements of individual ions. Ions dart about haphazardly, executing a random walk. By an analysis of one-dimensional random walk, a simple law can be derived (see Section 4.2.6) for the mean square distance-cc traversedby an ionin atimef. This is theEinstein-Smoluchowski equation... 

Fig. 24.8. Computational simulation analysis of conformational dynamics in T4 lysozyme enzymatic reaction, (a) Histograms of fopen calculated from a simulated single-molecule conformational change trajectory, assuming a multiple consecutive Poisson rate processes representing multiple ramdom walk steps, (b) Two-dimensional joint probability distributions <5 (tj, Tj+i) of adjacent pair fopen times. The distribution <5(ri, Ti+i) shows clearly a characteristic diagonal feature of memory effect in the topen, reflecting that a long topen time tends to be followed by a long one and a short fopen time tends to be followed by a short one...

The analysis of this model is similar to that of the well-known random-walk model, which was first developed to describe the random movement of molecules in an ideal gas. The only difference now is that for the freely jointed chain, each step is of equal length 1. To analyze the model one end of the chain may be fixed at the origin O of a three-dimensional rectangular coordinate system, as shown in Fig. A2.1(b), and the probability, P(x,y,z), of finding the other end within a small volume element dx.dy.dz at a particular point with coordinates x,y,z) may be calculated. Such calculation leads to an equation of the form (Young and Lovell, 1990) ... 

Let us return to the analysis of Brownian motion. For simplicity we begin by considering the continuous one-dimensional translational Brownian motion as represented by a one-dimensional random walk problem. The probability of a displacement between x and x + dx after n random steps of length I is given by the Gaussian distribution... 

The starting point for a molecular theory of diffusion is the analysis of a random walk of an atom or molecule. In Chapter 3, we dealt with the possible shapes of a one-dimensional polymer chain Eq. (3.13) gave the number of distinguishable chains of end-to-end length r as... 

Note that Equations (5.3)-(5.6) have been derived for the case of the onedimensional random walk. This is because these equations will be used later in analysis of sedimentation under gravity where motion only in one direction (one dimension) is of interest. Only motion of particles in the direction of the applied gravitational force field is of interest in sedimentation lateral motion in the other two orthogonal directions is not. In the case of the three-dimensional random walk, the analogy to Equation (5.3) would be L = V6af. 

Peterson, S.C. and Noble, P.B., A two-dimensional random-walk analysis of human granulocyte movement. Biophys. J., 1972,12 1048-1055. 

It optimizes an experimental study. In some cases, dimensional analysis reveals that certain parameters are not pertinent. The period of a pendulum is independent of mass, as is the velocity of walking. 

It can be rightly said that The reciprocal lattice is as important in crystal structure analysis as the walking stick of a blind man moving in a narrow lane having frequent turns. It is extremely difficult if not impossible to picture the different intersecting crystal planes satisfying the Bragg s reflection in three-dimensional lattice from the two-dimensional array of spots or lines. 

In Section 2.0 we examine trapping experiments in the aryl vinyl polymers, for which the trap is an excimer forming site (EFS). A one dimensional random walk model is used for analysis of the transient and photostationary fluorescence in dilute solution. In a second study, the three-dimensional results of LAF [5] are applied to trapping experiments in pure solid films of aryl vinyl polymers. 

The objective of this section is to present a quantitative analysis of the photostationary state fluorescence of miscible and immiscible PS/PVME blends. In Section 4.1 we develop the onedimensional random walk model that is used in conjunction with rotational isomeric state calculations to analyze low concentration miscible blends. In Section 4.2 we treat miscible blends having high PS concentration using a spatially periodic three-dimensional random walk model. Finally, in Section 4.3 we present a simple two phase morphological model and demonstrate how it may be used to monitor phase separation kinetics. 

For diffusion in a porous material with randomly oriented pores, the diffusion process is often assumed to be random as well. Thus, independence of pore to pore transit rates is an important assumption in the analysis of Pismen discussed above. Since the pore structure is fixed, however, this assumption is not necessarily correct. In the simplest case, consider one dimensional diffusion in a continuum and In a material with fixed porous geometry [26]. For a random walk in a continuum, the diffusing molecule loses all history at each step. The molecule moves either to the left or to the right at each step both events have probability 1/2. All possible coordinate points in the material are accessible. In a porous material the medium as well as the molecule is random, but the geometry of the material is fixed. For diffusion in a porous material, the molecular movement is no longer completely random, but is determined by the fixed geometry of the porous material. In fact, movement of molecules on a onedimensional lattice is completely deterministic. For materials of higher dimension, correlation of movement between pores in the medium must be considered [27]. 

The section is adapted from [Janvresse et al. (2005)], where the analysis in dimension (1 + 1) is restricted to symmetric random walk with finite variance and therefore to a = 1/2. But also the case of walks in dimension (1 + 2) and of (2 + 1) dimensional effective interface models are considered... 

Direct observation of molecular diffusion is the most powerful approach to evaluate the bilayer fluidity and molecular diffusivity. Recent advances in optics and CCD devices enable us to detect and track the diffusive motion of a single molecule with an optical microscope. Usually, a fluorescent dye, gold nanoparticle, or fluorescent microsphere is used to label the target molecule in order to visualize it in the microscope [31-33]. By tracking the diffusive motion of the labeled-molecule in an artificial lipid bilayer, random Brownian motion was clearly observed (Figure 13.3) [31]. As already mentioned, the artificial lipid bilayer can be treated as a two-dimensional fluid. Thus, an analysis for a two-dimensional random walk can be applied. Each trajectory observed on the microscope is then numerically analyzed by a simple relationship between the displacement, r, and time interval, T,... 

The random walk approach is based on the random-walk concept, which was originally apphed to the problem of diffusion and later adopted by Flory [3] to deduce the conformations of macromolecules in solution. The earliest analysis was by Simha et al. [4], who neglected volume effects and treated the polymer as a random walk. Basically, the solution was represented by a three-dimensional lattice. 

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