Section 43–1 Collisions between molecules

Diffusion of ions / Mean collision time / Collision & Survival probability

 

In this section, Feynman discusses the concept of mean collision time, together with the related ideas of collision probability and survival probability, in the context of ions diffusing through a gas under a constant electric field. Historically, the mean collision time (or mean free time) in the kinetic theory of gases was understood as a consequence of the mean free path and the mean molecular speed. In the work of Maxwell and Boltzmann, it evolved into a statistical average characterizing the random motion of molecules. Drude gave the concept new importance in 1900 when he explained Ohm's law in metals: electrons move freely between collisions with lattice ions, and their mean collision time (or relaxation time) became the key parameter linking microscopic collisions to macroscopic conductivity, diffusion, and mobility.

 

1. Diffusion of ions

“… we shall consider the diffusion of ions in a gas. Suppose that in a gas there is a relatively small concentration of ions—electrically charged molecules (Feynman et al., 1963).”

 

Definition: The diffusion of ions in a gas under a constant electric field is a transport process that combines random thermal motion with a net drift along the applied field. The ions undergo a random walk: collisions continually randomize their velocities, while the electric field accelerates them between collisions, producing a small average drift velocity superimposed on much larger microscopic molecular speeds. The gas is assumed to remain close to thermal equilibrium, so that the velocity distribution differs only slightly from the isotropic Maxwell–Boltzmann distribution. Collisions are further assumed to occur at a constant average rate, or equivalently a constant mean collision time. Under this (Poisson) collision model, ion transport can be characterized by a diffusion coefficient and mechanical mobility. Statistically, the motion involves two effects: diffusion, which spreads ions from higher to lower concentration, and drift, which directs a net motion along the electric force.

 

Idealizations: In deriving the equations for ions in a dilute gas, at least four idealizations are introduced.

1. Instantaneous collisions: The time between two successive collisions is much longer than the duration of a collision itself—meaning ions are relatively far apart.

2. Two-body collisions: The chance of three or more ions coming close to each other and interacting is negligible small compared to the chance of two ions doing so.

3. Classical particles: The ions are treated as classical particles whose de Broglie wavelengths are negligibly small compared to the mean free path and the characteristic dimensions of the system.

4. Weak electric field: The applied electric field is sufficiently weak that the drift velocity acquired between collisions is much smaller than the ions’ mean thermal speed. 

Under these idealizations, quantum effects can be ignored and the ions may be modeled as classical particles obeying classical kinetic theory.

 

2. Mean Collision time

“When we say that τ, the mean time between collisions, is one minute, we do not mean that all the collisions will occur at times separated by exactly one minute. A particular particle does not have a collision, wait one minute, and then have another collision. The times between successive collisions are quite variable (Feynman et al., 1963).”

 

A General Definition of Mean Collision Time

The mean collision time, t, is the average time that a particle (e.g., an ion or gas molecule) travels between successive collisions. In kinetic theory, collisions are assumed to occur randomly and independently at a constant average rate 1/t. It means that the probability of a collision during an infinitesimal time interval dt is dt/t. This assumption implies that a memoryless* collision process: the probability of a collision in the next instant does not depend on how much time has elapsed since the last collision. Under these conditions, collisions are described by a Poisson process, and the distribution of free-flight times is exponential. Most particles therefore collide after relatively short times, while a smaller fraction “survive” much longer than the average. Thus, t is not the actual time between successive collisions, but as the statistical mean of an ensemble of randomly distributed collision intervals. It provides a convenient measure of collision frequency in dilute gases and other near-equilibrium systems.

 

*Note: The memoryless property can be expressed mathematically as

P(T > s + t | T > s) = P(T > t) where T is the waiting time until the next collision.

 

Feynman-style explanation

In a sense, Feynman did not give a proper Feynman-style explanation of the mean collision time. So here’s an attempt:

 

I don’t have a simple definition of mean collision time. You might ask, “How long does a particle—an ion or a gas molecule—travel before it gets hit by something else?” Well, there isn’t any definite time. Sometimes it collides almost immediately; sometimes it wanders around for quite a while. The best we can do is talk about an average, and we call that the mean collision time, or τ.

Suppose we watch a particle for a very short time interval, dt. The chance of it collides with another particle during that interval is just dt/τ. That’s the whole assumption. It means collisions occur randomly, but with a definite average frequency, 1/τ.

 

Here’s the funny part: the particle has no memory of its past. It doesn’t matter if it just had a collision a microsecond ago or has been cruising for a long time—the chance of a collision in the next dt is exactly the same. You might think a particle that has gone a long time without colliding is somehow “due” for one, but nature doesn’t work that way. Every instant is a fresh start.

 

Because of this memoryless property, the times between collisions follow an exponential distribution. Most particles collide again after relatively short times, but a few lucky ones travel much longer than average before the next collision. If you collected all those flight times and average them, you would get τ—the mean collision time.

 

So τ is not the time between any particular pair of collisions. It is only a statistical average, a single number that summarizes a vast collection of random events. Yet that single number turns out to be extraordinarily useful. It helps us understand diffusion, electrical conduction, and many other transport phenomena in dilute gases and similar systems.

 

3. Collision & Survival probability

“If we wish the probability of no collision, P(t), we can get it by dividing N(t) by N₀, so P(t)=e^−t/τ.(43.8). Our result is: the probability that a particular molecule survives a time t without a collision is e^−t/τ, where τ is the mean time between collisions (Feynman et al., 1963).”

 

It is useful to distinguish between collision probability and survival probability—two complementary aspects of the same random collision process.

Collision probability concerns the likelihood that a particle encounters a collision in the next infinitesimal time interval. In the kinetic theory model, collisions occur randomly and independently at a constant average rate. If τ denotes the mean collision time, then the probability of a collision during an infinitesimal interval (dt) is:

P(collision in dt) = dt/τ.

Survival probability, by contrast, asks a different question: what is the probability that a particle remains collision-free up to time (t)? Since the collision probability per unit time is constant, the fraction of particles that “survives” (remains collision-free) decreases exponentially with time:

P(survive until t) = e^{-t/τ}.

Thus, collision probability and survival probability are complementary descriptions of the same Poisson process. The former focuses on the instantaneous likelihood of a collision; the latter describes the cumulative probability of “no collision” over time (See figure below).

 

A useful way to summarize the distinction is:

Collision probability → What is the chance of a collision occurring in the next moment? (Instantaneous)

Survival probability → What is the chance that no collision has occurred up to time t? (Cumulative, time-dependent)

Source: (Reif, 1965) 

During the lecture, Feynman used a bus analogy to illustrate the concept of mean collision time, but this was omitted by the editors. Another instructive analogy is radioactive decay: the collision probability corresponds to the instantaneous decay rate, while the survival probability is related to the fraction of undecayed nuclei remaining after time t. The exponential survival law follows directly from the assumption that the decay probability is constant. In a sense, collision probability is the fundamental assumption, whereas the survival probability is its statistical consequence.

 

In the audio recording of the lecture [07 min: 50 sec], Feynman says something like: “If buses run at random on schedule—which is usually what happens (laughter). Then, when you come out at a bus terminal and say, ‘I have to wait 15 minutes,’ this is only true if the buses are exactly 15 minutes apart… Well, the bus is not a good example because they are organized in time to some extent, but the collisions are completely disorganized…”

 

Here Feynman compares molecular collisions to waiting for a bus. He jokes that buses often arrive “at random,” so even if the average waiting time is 15 minutes, it does not mean buses arrive precisely every 15 minutes. Some may arrive almost immediately, while others may be delayed much longer. He quickly points out that buses are actually a poor example because they are governed by a schedule, whereas molecular collisions are assumed to occur completely at random. Nevertheless, the analogy captures an important idea: the mean collision time is not a fixed interval between successive collisions but a statistical average arising from a stochastic process. Just as a commuter cannot predict exactly when the next bus will arrive, we cannot predict when the next molecular collision will occur. Importantly, both processes are governed by a constant probability per unit time and therefore exhibit the same exponential model.

 

Key Takeaways:

In this section, Feynman examines the microscopic transport of ions drifting through a gas under the influence of a uniform electric field. His analysis provides a fundamental statistical description of molecular transport by reducing the complex dynamics of countless collisions to a simple probabilistic model. At the heart of this framework are three closely related concepts: the mean collision time, collision probabilities, and survival probabilities.

 

1. The Constant Collision Rate: The Memoryless Process

Feynman begins with the assumption that collisions occur randomly and independently at a constant average rate. If t denotes the mean collision time, then the probability that an ion encounters a collision in an infinitesimal time interval (dt) is: P(dt) = dt/t.

The Key Insight: The collision process is memoryless. An ion that has moved freely for a relatively long time without colliding is no more likely to collide in the next moment than an ion that has just encountered a collision. The parameter (t) is therefore not a fixed interval between collisions, but a statistical average characterizing the overage collision frequency.

 

2. The Exponential Decay of Survival Probability

To transition from an infinitesimal step (dt) to a timeline (t), we may ask: “What is the probability P(t) that an ion "survives" up to time t without a single collision?”  By analyzing how this probability changes over a small change P(t + dt) = P(t)(1 - dt/t), Feynman sets up a differential equation that yields a classic exponential decay: P(t) = e^{-t/t}

The Key Insight: At t = 0, the survival probability is 1(=100%). As time increases, the fraction of ions that are collision-free decreases exponentially. After one mean collision time (t = t), the fraction of ions remain collision-free is only e^{-1}» 37%.

 

The Moral of the Lesson: Aerosol Dynamics in Public Spaces

The diffusion of viral aerosols (or smoke particles) in a public restroom provides a compelling, real-world application of kinetic theory and statistical mechanics. By analyzing how these particles behave, we can make informed decisions to improve public health.

 

1. The Gas Molecule Analogy

After a toilet is flushed, the water jet generates aerosol droplets that become suspended in the air. These droplets behave like gas molecules: they undergo random motion driven by continual collisions with air molecules and are transported within the toilet by diffusion and air currents.

 

2. Mean Residence Time and Airborne Survival Probability

An airborne particle remains suspended only for a finite time before being removed by one of several mechanisms: gravitational settling, surface deposition, or extraction via the ventilation system. If these removal processes operate at an approximately constant average rate, the probability that a particle remains airborne after a time (t) is

P(t) = e^{-t/τ}

where τ is the mean residence time of the particle in the room. More complex models exist (for example, Srinivasan et al., 2021), but this exponential approximation is often sufficient for near‑equilibrium conditions.

 

3. Constant Hazard rate

When the ventilation rate is steady and environmental conditions remains approximately constant, particle removal can be modeled by a constant hazard rate (probability per unit time), denoted by 1/τ. This quantity is analogous to the collision frequency used to describe transport phenomena in dilute gases.

 

4. Exponential decay of Particle Concentration

Because each particle has a constant probability of removal per unit time, the concentration of infectious or harmful particles decreases exponentially:

C(t) = C₀e^{-kt}

where C₀ is the initial concentration and k is the total clearance rate.

The clearance rate may include contributions from: mechanical ventilation, natural air exchange,  surface deposition, gravitational settling, and biological inactivation.

 

5. Application to Infection Risk

Wearing a protective mask in a public restroom is a practical application of collision‑based, probabilistic thinking. A mask acts as a physical barrier that intercepts moving aerosols before they reach your respiratory tract:

Fibers as Collision Targets: When breathing through a properly fitted mask (e.g., N95 or surgical mask), the network of crisscrossing fibers acts as a high-density of collision targets.

Capture Probability: The cumulative probability of capture depends on the droplet’s size, mask’s filtration efficiency, and airflow conditions. In statistical terms, a mask decreases the probability that a particle successfully survives its journey from the environment to the lungs.

 

Key Tips for Daily Life

Close the lid before flushing: Whenever possible, close the toilet lid prior to activating the flush. The lid acts as a physical barrier that helps reduce the upward transport of aerosolized droplets, thereby limiting harmful bacteria and viruses from being atomized into the air.

Avoid Smoking or Vaping Indoors: Smoking or vaping release large numbers of fine airborne particles that can remain suspended for extended periods in poorly ventilated spaces. These particles can be inhaled by both the user and subsequent occupants, increasing the risk of respiratory illnesses and cancer.

Use Masks and Ventilation together: Ventilation increases the clearance rate (k), while masks reduce the probability that remaining airborne particles are inhaled. The combination of effective ventilation and well-fitted masks provides a better defense against airborne exposure. From the perspective of kinetic theory, this combined approach simultaneously reduces both the concentration of hazardous particles in the environment and the probability of an encounter between a particle and the respiratory system. In other words, it lowers both the collision rate and the total exposure time—two key factors that determine risks.

 

Review Questions:

1. Explain why the velocity distribution of ions in a weak electric field is assumed to deviate only slightly from the Maxwell distribution. Which physical assumption justifies this?

2. Explain what it means for the collision process to be "memoryless." Why does this property lead to an exponential distribution of free-flight times? Provide a simple analogy (e.g., radioactive decay or waiting for a bus) to illustrate the idea.

3. Explain why the collision probability is considered the "fundamental assumption" while the survival probability is a "statistical consequence." Explain how are the two probabilities complementary?

 

References:

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Maxwell, J. C. (1860). II. Illustrations of the dynamical theory of gases. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 20(130), 21-37.

Reif, F. (1965). Fundamentals of statistical and thermal physics. McGraw-Hill.

Srinivasan, A., Krishan, J., Bathula, S., & Mayya, Y. S. (2021). Modeling the viral load dependence of residence times of virus‐laden droplets from COVID‐19‐infected subjects in indoor environments. Indoor Air, 31(6), 1786-1797.