Using the Diffusion Coefficient to Estimate Crime Displacement
A Hypothetical Model to Changing Proactive Policing
“I can calculate the movement of the stars, but not the madness of men.” A quote attributed to famous physicist Sir Isaac Newton representative of the complexity found in scrutinizing human behavior. Although a renowned scientist and forward thinker of his, Newton made this observation after losing a hefty investment. Newton is not alone in his feelings. Long have scientists toiled with the labyrinth of mind, trying to understand in the least bit what drives us to “do.” Famous experiments such as the Milgram shock study totally altered our thoughts about reason in the human mind. Totally ordinary people, given a situation where there is an air of uncertainty and authority simultaneously were made to electrocute someone “to death” in a matter of minutes. Prior to the experiment, Milgram and his compatriots thought to themselves that only select few people would see the demonstration through to the point of no return.
Just one of many examples as to how the human mind and behavior is endlessly revealing yet so well encrypted; criminologists have been no stranger to the struggle. As I think of new ways to understand what creates the “motivated offender,” I think to myself “why can’t we define movement?” What is preventing us from understanding how or why or why not people move en masse? It would seem as though we would have every bit of the puzzle already. We sit and ponder the intricacies of systems both natural (tectonic plates, precipitation, the carbon cycle, planetary orbit, elementary particles) and those we have created (government, law, economy, infrastructure, computers) and yet we don’t know, still, hardly why anyone chooses to offend.
Of course we know on a case by case basis. In a courtroom a judge and jury and plaintiff and defendant ponder the evidence and make the best possible judgment as to who deserves what and why (most of the time). We also know from classical experiments in criminology some of the general motivators of crime. Routine activities theory clearly states that certain crimes are liable to happen when a motivated offender encounters a fitting target given the lack of a capable guardian. Strain theory finds that people may tend to commit crime more when they are strained by social, economic, or educational factors. All such theories are great and well proven within the social sciences, but they are not guaranteed.
This, I believe, arises when there is no set standard. Yes, there are three basic components to routine activities, but this is not always the case. People would not rob banks, a place where there is high levels of security and relatively low success rates, if routine activities always held true. Strain theory is certainly a key component in criminogenic systems, but if it always held true, it may be a worthy use of time to put all disadvantaged peoples in jail for they will inevitably offend.
Considering this, I believe it is time to start toward the formation of a middle ground. A time to create standardized, quantifiable systems that will have the capability to estimate probability in crime. While classical theory has largely remained nonquantifiable (we are unable to measure “strain” or define a “routine activity”), I draw from core concepts in physics for inspiration. Mainly, I will propose an adaptation and potential use of the diffusion coefficient as a method of estimating actual human movement in criminogenic systems.
Given above is the well known Stokes - Einstein - Sutherland equation. D in this equation is representative of the diffusion coefficient. Specifically, this equation is tailored to represent the diffusion of particles through a liquid. This was originally used by these physicists as a means to explain Brownian motion; the random movement of observed particles in a medium (liquid or gas). The movement is caused by the continuous collisions of the liquid or gas particles into the larger, observed particles, causing a seemingly random motion.
Originally documented by Robert Brown (hence, Brownian), Einstein and his colleagues, maintaining faith in the existence of atoms, thought to apply the idea of a random walk to a particle. A random walk is a mathematical expression that explains through probability the random behavior in a mathematical space of any given thing. A stochastic process, it can be thought simply as understanding the probability that any one thing will make a given move in a given time and space. Once the movement is defined, it can be plotted as a curve in plane.
This discovery was integral in demonstrating the existence of atoms at a time when physicists bore split opinions on the nature of what is happening at unobservable levels. Affirmed by Jean Perrin when he measured particle displacement in line with Einstein’s prediction,
, he was able to plug these observations directly into the diffusion expression, which yielded a precise estimation of Boltzmann’s constant. From there, Avogadro’s number (the number of particles in 1 mole of a substance) and, thus, the mass of a single atom could be calculated.
Einstein’s breakthrough in quantum physics was integral to our current understanding of the world. As one can imagine, many lifechanging discoveries have followed since. I believe, however, it is time to rethink aspects of criminology following in the footsteps of Einstein.
Let’s look back at his original equation:
Whereby:
D = the diffusion coefficient
kB = the Boltzmann constant
T = absolute temperature
η = dynamic viscosity
r = Stokes radius
The diffusion coefficient is what we are most interested in, of course. I would like to hypothesize reapplying this concept to visualize the movement of crime or criminal activity across space and time. In this case, let’s look at it from the perspective of crime displacement in response to policing activity. Now, D, when the expression is computed given a set of variables, will yield a number that can be related to how crime or criminal activity will move. Before we can do that, though, let’s define the rest. Boltzmann’s constant is a universal number that draws a connection between temperature and energy at the atomic scale. With this number, we can explain why gases expand or contract at given temperatures and so on.
Now, the Boltzmann constant is used to estimate the physics of particles or atoms at a very small scale. Unfortunately, it will be unruly to apply this directly to individuals in a macro-scale, criminogenic system. Instead, let’s substitute this for the Boltzmann entropy equation:
Whereby W is the possible number of social states, this is commonly utilized in information theory to model decision making or human behavior. S, then, will serve as a variation of the Boltzmann constant. T serving as absolute temperature of system, again, is beneficial to understand micro-scale events where the temperature directly effects the velocity of a particle (higher temperature entails higher velocity, lower vice versa).
In studies on proactive and hotpot policing, it is often said that the treatment of police to a certain hotspot is the “dosage.” This is generally quantified as the amount of officer, patrol cars, or the like that show up to a crime hotspot. In this case, T could be replaced directly as, for instance, P: the given police presence as the treatment. In this case, it would likely be best to define it as the amount of police cars that are sent to a given area or hotspot. Should P be too low, the presence will not be effective, yielding a lower D. Should it be too high, the diffusion will become “turbulent,” crime dispersing uncontrollably. Hypothetically, other factors that influence the distribution of crime such as opportunity, social disorganization, or poor economic status in an area could be applied equally.
Concerning η, the dynamic viscosity of the system. Traditionally, this is representative of the viscosity, or fluidity, of a liquid or gas in a micro-scale system. This is calculated with the following:
Whereby:
ρ = mass density of the gas
λ = mean free path of molecules
v = average molecular velocity
Specifically, this is the kinetic theory of gas approximation. This should serve well for a criminogenic system as crime, at a macro-scale, exists in systems where by there is a mean free path of ways it can go (of molecules). Given a city, you could map out the city and theoretically account for every possible way that crime can diffuse λ. λ may be founded as the “criminogenic mean free path,” based on a variation of the kinetic theory mean free path equation given below.
Instead of thinking of this in traditional terms, of course, we will adapt it for purposes of measuring crime diffusion whereby:
λ = criminogenic mean free path (average distance before encountering a crime event)
σ = criminogenic cross-section (a measure of how "visible" or "impactful" a criminal interaction is—higher for gang territories or surveillance-limited areas)
Factor radical 2 is necessary given this will measure random movement in 2 dimensions. σ is measured given the area of impact of the system. How far the reach of their crime spreads. This is designed more to think of how the average observer will interact with the criminogenic system, not law enforcement. Hence, the average distance before encountering crime.
ρ, the mass density, in the current model will be analogous to crime density, or the reported number of crimes in the given area. This will be done by dividing the number of total crimes in the area by the population, giving us the base crime rate. The crime rate will be given over the spatial area of where the population resides.
Whereby:
N = crime rate
A = area
v may be a more abstract way of accounting for diffusion, but, if given the known circumstances of how crime may diffuse with all the possible ways where it can diffuse, we could estimate velocity should crime diffuse. Whether it be on foot, car, bicycle, we can calculate where crime may go as the individuals bearing the quality of “crime” choose to displace in the bounded system (city, town, neighborhood).
Concerning the rest of the bottom of the equation,
, justification is needed. This part is derived from Stokes’ law of friction:
where the velocity is it’s own distinct part. Recall that as part of η, the velocity has already been incorporated. Now, 6 itself is derived as from the mathematical derivation of Stokes’ law which entails that diffusion is observed and accounted for in three dimensions. Think of a cube with 6 sides. It is a direct function and product of a square in two dimensions having 4 sides where as, when it is raised to the third dimension, it then has 6. This is also why π is stated in the equation. r itself is representative of the radius of a spherical particle in space which we are observing. π appears naturally as it linked with the geometry of a sphere. It’s presence has a pivotal role in calculating fluid resistance in dynamics. This all governs r, the radius of a particle.
Now, r is the last piece of the equation which deserves justification or relation. Well, in micro-scale systems, the particles with larger r will inherently have greater resistance to movement. I suppose this can be sufficiently interpreted in a criminogenic system as the area of effect that the system has. Does the system effect or operate in one household/building, a neighborhood, a district, and so on? While this would be difficult to calculate, I think that, if have been observing a criminogenic system for long enough, it should be reasonable to know the minimum number of people it has effected. Or the minimum area? r could also be representative of a social network of influence.
Given all of this, our equation that yields a number representative of how crime may diffuse in response to police presence is as follows:
Whereby:
S = Boltzmann’s entropy formula
P = Police dosage (substitution of absolute temperature T)
η = Resistance to displacement (relative replacement to dynamic viscosity)
r = radius of effect (relative replacement to particle radius)
The ideal application to test this would be to attain data on crime displacement in reaction to hot spot or proactive policing that has occurred over a multitude of years. Ideally, this would be centralized on a city or town, somewhere big enough with enough resources to track crime displacement effectively. For each part of the city or town (neighborhood, block, tract), find the appropriate measures that would fit with our newly refreshed model. Of course, for each specific use case, you would need to know the apportioned dosage for hot spot policing to satisfy the conditions of P. Given the construction of the selected area, you would be able to determine W in S, each possible states of crime diffusion. This would be, given the nature of what crime is being observed, where the crime could possibly go in real time; it would be largely dependent on the construction of the streets, alleyways, public transportation, where parking lots are, shaded areas, parks, etc. It is a lot to keep track of for each case, but not impossible. Services like Google Maps have already extensively documented everything in our cities and towns; the streets, buildings, addresses, alleyways.
Resistance to displacement, η, and r will go hand in hand estimating every reason why or why not it may be possible for crime to diffuse or displace. η being the function directly correlated to the size constriction. One offender may be able to evade law enforcement contact easily while a network of individuals is more likely to have contact. r, as just covered, is the radius of effect of the criminogenic system. Again, this increases the likelihood that a criminogenic system cannot displace itself or diffuse. It will be based on prior known data of who or what the system affects.
Once you have all proper variables calculated over a given time span (which a modern computer would be able to do in a matter of moments), you could produce the diffusion coefficient of crime given a certain set of conditions at any time in space. This number could be related to the state of crime displacement at any time. Theoretically, you would be able to utilize the data given on crime displacement in relation to hot spot policing tactics and, with a Bayesian inference technique, apply what I have stated. This could effectively estimate the diffusion and displacement of crime given hot spot policing in the future.
To take this a step further, consider on an individualistic basis. Risk-needs assessment tools have long stood as a means of calculating an individuals likelihood of reoffending. My girlfriend, Cecilia Nguyen, who is super wonderful, struck this idea. When thinking of how an individual convict might behave when leaving incarceration, we consider all elements of their behavior and how likely they are offend. Readjusting the equation, we could consider their likelihood to recidivate as the viscosity or as an alternative to police dosage. This may yield a, well less descriptive account for diffusion or likelihood to recidivate. It would tell us something nonetheless. This would require another lengthy discussion, but it is worth considering. While I cannot dedicate full thought to this now, Cecilia still helped with much of this paper. Thank you.
If you read all of this, thanks. I had this done a while ago but got hung up on some other things. I have somewhat rushed to send it out, but I hope it makes sense. I see this as a newer way to describe criminal behavior. There is other literature in the field of applied mathematics using similar processes of Lévy flight to estimate movement. I take up issue with this as most of the literature relies upon the cops-on-dots model of proactive policing. However, if one is to dive into the literature, one will find that most success with cops-on-dots in found in the apprehension of pedestrians, not those who are making a Lévy flight. Oh these applied mathematicians, would they ever read a paper on the social sciences? Anyway, please like and subscribe (it’s free).
