# Regimes of Particulate and Wave-like Behavior

• Vrbic
In summary, the conversation discusses a gamma-ray burster, an astrophysical object that emits gamma rays with a total energy per unit area of approximately 10^-6 ergs/cm^2 and lasts for about one second. The mean occupation number of the burst's photon states is estimated to be approximately 2x10^-20, assuming classical, distinguishable particles. The occupation number will not change as the photons propagate from the source to Earth.
Vrbic

## Homework Statement

A gamma-ray burster is an astrophysical object (probably a fireball of hot gas exploding outward from the vicinity of a newborn black hole or colliding neutron stars or colliding neutron star and black hole) at a cosmological distance from Earth (∼ ##10^{10}## light years). The fireball emits gamma rays with individual photon energies as measured at Earth E ∼ 100 keV. These photons arive at Earth in a burst whose total energy per unit area is roughly e=##10^{−6} ergs/cm^2## and that lasts about one second. Assume the diameter of the emitting surface as seen from Earth is D∼ 1000 km and there is no absorption along the route to earth. Make a rough estimate of the mean occupation number of the burst’s photon states. Your answer should be in the region η << 1, so the photons behave like classical, distinguishable particles. Will the occupation number change as the photons propagate from the source to earth?

## Homework Equations

##n=\frac{c^2 I_{\nu}}{h^4 \nu^3}##...distribution function
##E=h\nu##...energy of one photon
##I_{\nu}=\frac{dE}{dAdtd\nu d\Omega}## intesity = energy per unit area unit time unit frequency unit solid angle
##\eta=\frac{h^3}{g_s}n## ...mean occupation number, where ##g_s=1## for photons

## The Attempt at a Solution

I believe it is easy, I have only one uncertainty, I suppose that ##I_{\nu}\sim e/(D/2)^2=10^{−6} ergs/cm^2/(D/2)^2##. Is it good approximation?

Assuming that, I get:##n=\frac{e}{h^4 \nu^3 D^2}## ##\eta=\frac{h^3}{g_s}n=\frac{eh^2}{g_s \nu^3 D^2}## ##\eta=\frac{10^{−6} ergs/cm^2 h^2}{100 keV^3 1000 km^2}≈2×10^{−20}##No, the occupation number will not change as the photons propagate from the source to Earth.

## 1. What is the difference between particulate and wave-like behavior?

Particulate behavior refers to the movement of individual particles, while wave-like behavior refers to the movement of a group of particles as a wave. Particles exhibit particulate behavior when they maintain their individual identities and interact with each other through collisions. In contrast, particles exhibit wave-like behavior when they move in a coordinated manner, such as in a wave or a current.

## 2. How do particles transition from particulate to wave-like behavior?

Particles transition from particulate to wave-like behavior when they are close enough to each other to influence each other's movement. This can happen through attractive or repulsive forces, such as electromagnetic or gravitational forces. As particles interact and become more closely packed, they can exhibit wave-like behavior.

## 3. Can a single particle exhibit both particulate and wave-like behavior?

Yes, a single particle can exhibit both particulate and wave-like behavior, depending on its environment. For example, a single particle in a vacuum will exhibit primarily particulate behavior, while a single particle in a dense medium, such as water, may exhibit more wave-like behavior due to interactions with other particles.

## 4. What factors influence the behavior of particles in a system?

The behavior of particles in a system can be influenced by a variety of factors, including the properties of the particles themselves (such as size, shape, and composition), the environment in which the particles are located (such as temperature and pressure), and the interactions between particles (such as attractive or repulsive forces).

## 5. What are some real-world examples of systems exhibiting both particulate and wave-like behavior?

Examples of systems exhibiting both particulate and wave-like behavior include water waves, sound waves, and electromagnetic radiation. In these systems, individual particles (such as water molecules, air molecules, or photons) exhibit particulate behavior, while the overall movement of the particles can be described as a wave. Other examples include traffic flow, crowd behavior, and the movement of cells in biological systems.

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