Alternatively to the illustrative but mechanical stone-valley system, we may also think of x as denoting a psychological variable, e.g., “positive affect.” A given individual may possess a specific, most comfortable level of affectivity, so that— once outside this zone of comfort—a deterministic psychological force Ks will become active and drive the psychological system back toward this zone of comfort.
In this case, we are dealing not with a gravitational potential as in mechanics but an
“affective potential,”i.e., a psychological potential.
As a consequence, the velocityvsalong the slope of the stone (or the“slope”of the individual’s affectivity) is proportional to the acting forceKsalong the slope. For our purpose it suffices to put directlyvs¼Ks.
Furthermore, in the following we need to consider the change of x, i.e., the projection of velocityvsand forceKsonto thex-axis,vs!v,Ks!K(Fig.4.3).
We assume that the forceKdepends on the size of the slope that itself changes with the variablex, so thatK¼K(x) and thus also
v xð Þ ¼K xð Þ ð4:2Þ In which way doesKdepend onx? For largex, the slope becomes steep and thus Klarge. Forx¼0,K(x) vanishes,K(0)¼0. The simplest assumption isK(x)¼ kx, wherekis a positive constant (actually, this relationship can be derived exactly for a parabola—see the legend of Fig.4.3). The minus sign is needed because the stone moves to the left for positivexand to the right for negativex. After these prepara- tions we may“translate”our deterministic description into a probabilistic one (see Info-Box4.1). Our treatment still concerns thedeterministicdynamics of a system in a forcefield (described by a potential function)—we just model this dynamics using probabilistic terminology.
In our presentation, we have considered the motion of an individual stone. To better understand the following, however, we may consider an ensemble of many stones, each of which is moving deterministically in its own landscape, and where all landscapes possess precisely the same shape. Or, respectively, one may imagine many different but independent situations, i.e., instantiations of one and the same stone. The distribution over an ensemble of many stones can be treated analogously to the distribution over an ensemble of situations of one stone.
To better understand the notion of probability, just think of a “percentage of stones”—e.g., what percentage of stones can be found at positionx(in region Δ)? And furthermore, how does the percentage of stones change in the course of time (again, mind the analogy to a “percentage of situations”)? As the stones are all moving, some will enter the considered region; others will leave it (Fig.4.4).
This description leads to master equations that are generally such equations which describe changes of a state of a variable by the difference between an influx to that state and an outflux from that state. For example, in population dynamics, the change (i.e., the decline or growth) of the size of a population can be simply defined
4.3 Deterministic Processes: Causation 45
by the difference between the birth rate and the mortality rate. The same logic is applied in the probabilistic description of the change of a state variablexof a system.
The relevant measure of change is the probabilityPthat the system occupies position xat timet, i.e.,P(x;t) (see Info-Box4.1).
Info-Box 4.1
Probability P depends on position x and time t, i.e., P(x;t). We consider adjacent regions. The change ofP within a time intervaldtat position xis given by
P xð ;tþdtÞ P xð Þ;t ð4:3Þ and its rate of change by
1
dtðP xð ;tþdtÞ P xð Þ;t Þ ð4:4Þ This change is caused by:
1. An influx from the region atx+dx. The size of this influx is given by the percentage of system states atx+dxtimes their speed of change atx+dx, i.e., byv(x+dx). This speed can be expressed byv(xþdx)¼ k(xþdx).
Since the influx must be positive, we drop the minus sign so that altogether influx¼k xð þdxÞP xð þdx;tÞ ð4:5Þ
2. In complete analogy, the outflux is given by
outflux¼ kx P xð þdx;tÞ ð4:6Þ (note the minus sign!)
All this happens within a time interval dt and a corresponding length interval dx. Thus we obtain a master equation for the rate of change of P(x;t) in the course of time (4.4):
¼ 1
dxðinfluxoutfluxÞ ð4:7Þ (continued)
Info-Box 4.1 (continued)
(continued)
Info-Box 4.1 (continued)
In the case of a continuous time variable, (4.4) becomes dP xð Þ;t
dt
Ifx is a discrete variable (i.e., is countable, e.g., measured by a scale), Eq. (4.7) with (4.5) and (4.6) is an example of a master equation. If x is continuous, the limitdx!0 is taken and (4.7), (4.5), and (4.6) become
dP tð Þ dt ¼ d
dxðkxP xð Þ;t Þ ð4:8Þ This is the deterministic part of aFokker-Planck equation.
To visualize how the probability changes in the course of time in the specific case of mechanics, think of an ensemble of stones, where each stone is moving in its own valley.
For illustration we consider an initial state where according to Fig.4.6the stones initially occupy the positions betweenx1andx2with equal probabilityP(x;t)¼const. To see what happens when we let the stones slide down their hills, look at Fig.4.5. At position x, a stone slides down with some velocityvthat depends on its positionx(Info-Box4.2).
When we compare the velocities of the stones atx1andx2, respectively, according to Info-Box4.2, the velocity atx2is larger than that atx1(Fig.4.6). This means that the probability distribution shrinks more and more in the course of time, so that the width of the distribution becomes smaller. Since the stones cannot vanish, the total probability is preserved (Fig. 4.7), and correspondingly the shaded area is also preserved for all time. Our argumentation is valid for all kinds of probability distributions, Gaussian or non-Gaussian.
Fig. 4.4
4.3 Deterministic Processes: Causation 47
Info-Box 4.2
In the probabilistic description of a deterministic dynamics, we plot a proba- bilityPversus thex-axis (Fig.4.5). If the stone is with certainty at position x (within an error region Δ), we just draw a line. When the stone slides downhill, this line moves at a speed v(x) ¼ K(x) (cf. Eq. (4.2)) to a new position. Let us consider a situation where the stone can initially be found at positionsx1orx2¼x1+d(Fig.4.6). Sincex2is larger thanx1, the forceKatx2 is larger than that atx1so that the stone at x2 moves faster than atx1. The distancedshrinks in the course of time.
The shrinkage of probability distributions means that, quite generally, any probability distribution that is subject to a deterministic dynamics shrinks in the course of time (Fig.4.7). This shrinkage is an expression of the defining character- istic of an attractor—attractors have the property of compressing state space. In the basin of an attractor, state space volumes become smaller with time.
Fig. 4.6 Fig. 4.5