International Theoretical Physics Olympiad

Eine Maus, Siegfried, is stuck in a labyrinth. Siegfried was a house mouse of Albert Einstein, so it knows a bit of General Relativity. Because of that, Siegfried is afraid of black holes. Please help Siegfried overcome this fear, by calculating the probability that the labyrinth collapses into a black hole. Siegfried is smaller than we are, and can see smaller things than us, like extra dimensions. Hence, assume that Siegfried lives in $d$-dimensions. The labyrinth could be considered as a graph where each vertex has a valence $q$. The labyrinth is random, meaning that edges coming out of each vertex are uniformly distributed on an $S^{d-1}$-sphere. Each edge is weightless and has length $L$, and each vertex has mass $m$. What would you say about the same labyrinth, when the mass is uniformly distributed along each edge while the vertices are weightless?

Dangerous bacterial strains are able to adapt to antibiotic treatments by gene manipulation. Rather than sexual or asexual reproduction, which both involve the creation of new bacterial units to achieve genetic mutations, some bacteria are able to "infect" other living bacteria with foreign genetic code. Viral transfer of DNA is hypothesized to be a dominant contributor to the spread of antibiotic resistance.

Imagine lining a square petri dish (with side length $L$) with a linearly increasing concentration of an antibiotic (with slope $\kappa$), and a uniform distribution of food. Place an initial bacterial population on one boundary of the dish, where the antibiotic concentration is 0. Then, model the dynamics of the population, and calculate (either analytically or numerically):

1) The time it takes for the bacteria to reach the opposite boundary of the dish where the antibiotic is maximally concentrated.

2) The percentage of resistant bacteria in the population over time, and the distribution.

Assume the bacteria are identical, except for a small fraction of the initial population $\rho$ which has a mutation that makes it partially resistant to the antibiotic. In the absence of the antibiotic, the rate of reproduction of the resistant bacteria is slower by a small amount $\delta$, as the developed resistance has a metabolic cost. You may assume that the bacteria have a characteristic size $\ell \ll L$, that the bacteria do not move or overlap, and that a single resistant bacterium will infect its nearest neighbors with the resistance gene at a constant rate $r$. You will have to reason about the effect of the antibiotic on the rate of reproduction of the bacteria. With justification, you may assume that the decrease in the rate of reproduction is linear in the total amount of antibiotic in contact with a bacterium, with differing slopes $c_{nr}$ and $c_r$ between the populations (where $c_r < c_{nr}$).

Consider the following matrix path integral:

where the integration is over the elements of an $N \times N$ Hermitian matrix and $V(M)$ is an arbitrary potential. Such path integrals are understood to be dual to minimal string theories, whose worldsheet actions involve Liouville gravity with a minimal conformal field theory in the matter sector.

1) Take $V(M) = \frac{h}{N^2} (\text{Tr} M)^2$, where $h$ is a coupling constant. Calculate the probability distribution of a single eigenvalue of $M$, assuming $N$ is very large.

2) Compute the same distribution assuming $V(M) = \frac{g}{N} \text{Tr} M^{2k} + h\left[\exp\left(\frac{1}{N} \text{Tr} M^{2k}\right) - 1\right]$. (You will receive partial credit if you solve for specific values of $k$).

There is a common illustration of how gravitational forces arise within the theory of general relativity (GR). Consider a trampoline, a stretched piece of elastic fabric, with a heavy ball placed in its center. If smaller balls are placed on the fabric of this toy universe, they will move along trajectories resembling ones in the gravitational field of a heavy object. In this problem you are suggested to study the system described above and answer the following questions:

0) Imagine that a coordinate grid was drawn on the flat trampoline. Before being deformed by the heavy ball, the metric was flat. Describe the metric after the deformation. Does it look like a Schwarzschild solution?

1) What are the possible trajectories of a ball on the trampoline deformed by a heavy ball at the center? Assume that the balls have finite radii, that there is friction, and that you may work in the probe limit for now.

2) What are the leading effects of the boundary shape on the ball trajectories? Assume that the boundary of the trampoline can be drawn on a flat piece of paper.

3) In GR, two orbiting objects will radiate gravitational waves and eventually collide. Is there an analogous radiation emitted by balls moving on the trampoline and propagating in its fabric? If so, estimate the intensity of this radiation.

The animal world of the Flatland (a reference to the novel by E.A. Abbott about a 2D country) is quite versatile. Take, for example, the myriapods. These creatures consist of a body with $N$ segments, $N \gg 1$. Each segment is a piece of a straight line of length $\delta$. In the middle of each segment there is one leg, a straight interval of length $a/2$, such that $N\delta \gg a/2 \gg \delta$. Legs are always orthogonal to the segments and can go from one side of the segment to another if needed. In Flatland, myriapods can pass through each other like ghosts.

Myriapods prefer to sit straight. Their happiness level depends on the angle $\theta_{i,i+1}$ between the consecutive segments as $\sum_i \kappa \cos \theta_{i,i+1}/\delta$. They are also very extroverted animals. Namely, they like to hold each others' legs so that the ends of their legs coincide, forming a straight piece of length $a$. Their happiness level has a contribution proportional to the number of held legs with constant $\mu\delta$. Assume $N\delta \gg \sqrt{\kappa/\mu} \gg \delta$.

Consider a system of two myriapods. Obviously, the state of maximal happiness is when both myriapods are perfectly straight and hold all legs. However, due to mistakes upon their meeting they may end up in a metastable state, with the happiness level lower than a maximal possible, but such that the transition from this state to the state of maximal happiness would require an initial decrease in happiness by an amount proportional to the total number of segments.

1) Classify the possible metastable states (defects). When analyzing defects which are observable even when $a = 0$, you may simplify and set $a = 0$. Otherwise, keep $a$ nonzero.

2) Consider pure defects of each type, such that effects of other defect types are not present. How many independent parameters does each defect have? Do these parameters completely determine the shape of the myriapods?

3) In each class of defect, determine the angle between directions of the myriapods at infinity, both on the left and on the right of the defect, as a function of the defect parameters, as well as $\kappa$ and $\mu$.

4) One class of defects demonstrates a phase transition when parameters of the problem are varied while the parameters of the defects are fixed. Identify this class, as well as the behavior in the vicinity of the phase transition. You may solve this part of the problem numerically.

Most dark matter direct detection experiments involve massive collaboration efforts surrounding large-scale engineering projects and cutting edge instrumentation. Your challenge is to design a dark matter detector using much more rudimentary materials, but unbound by real-world financial or practical constraints.

1) You might remember childhood activities in which you were given paper, scissors, glue, and creative freedom. Perhaps you even yearn to relive those times. Well, good news! Your government has an infinite surplus of paper and glue, and has passed it off to you, free of charge. Use these resources to construct a detector of self-interacting dark matter. Because you have saved so much money on the apparatus itself, your detector is not required to be on Earth, or anchored to any object for that matter. You may also assume convenient qualities for your "paper" and the environment of your detector, to modify aspects such as density, combustion point, temperature, etc. within reasonability.

2) Now that you have a potential method of detection, what physical characteristics of your detector can be optimized? Is there an ideal volume or density of your detector which will maximize the rate of detected events?

In principle, boiling water can source gravitational waves. State why that is, and describe how gravitational waves may be produced in a teapot with boiling water. Estimate the peak of the power spectrum of gravitational waves produced in this process. Make and justify whatever assumptions you find reasonable, and always cite your sources.

Consider a free, massless scalar field φ in two-dimensional Minkowski spacetime. Show that the action is invariant under the transformation rule

φ → φ + δφ ≡ φ − ε(z)∂φ − ε̄(z̄)∂̄φ

Where z and z̄ are the so-called "lightcone coordinates" in two dimensions, ∂ and ∂̄ are their corresponding derivatives, and ε(z) and ε̄(z̄) are small but arbitrary functions of the indicated coordinates. Next, for a small parameter α consider the following Lagrangian density, as an extension of the free scalar:

L = ∂φ∂̄φ + α(∂φ∂̄φ)²

Is it possible to add O(α) contributions to δφ which are first-order in φ-derivatives and which preserve this invariance?

Write down a spacetime metric corresponding to a classical black hole system in an infinite number of spacial dimensions. Using your result, generate what is known as a multi-center black hole geometry. More specifically, describe an infinite-dimensional spacetime which has an arbitrary number of black holes interspersed at arbitrary locations.

Now relax the limit of infinite dimensions and consider this spacetime in a large enough number of dimensions d so that the geometry is nearly identical. Distribute a number density ρ of these black holes approximately uniformly across some d-sphere of finite radius R. Analytically or numerically, model the density of a diffusive gas of massive particles as it moves through this spacetime, provided the initial density of the gas is Gaussian-distributed with width ℓ ≪ R and a small total mass M.

When spherical air bubbles are released underwater, they drift toward the surface with a velocity which depends on their depth and their volume, among other physical criteria.

1. Model such a spherical bubble and calculate its upward drift velocity. Then, modify your model and compute the same quantity for a bubble of genus 1 – a "bubble ring." Assume the plane of the ring is approximately parallel to the water's surface.

2. Is it possible to make any comments (qualitative or quantitative) on the drift velocity of higher-genus bubbles?

3. Dolphins are known to play with each other by forming and manipulating stable bubble rings. How can the statement of this problem be modified so that the ring sits in a stable configuration underwater?

Two actively hostile high-altitude balloon experiments aim to launch a weather balloon. One of the experiments succeeds, and, in unusually opportune windowless conditions, the balloon begins to rise vertically upward. The payload attached to the balloon ensures it does not rotate. When the other group's PI sees the balloon, in a rage of uncontrollable jealousy, he shoots an air dart at the balloon, making a small spherical opening on its surface.

1. Predict the balloon's trajectory. Assume that initially, the thrust is much larger than drag, and that the balloon is low enough so that the surrounding air density does not vary appreciably as the balloon descends.

2. The other experiment still wants to save their balloon and their precious data. Can they also shoot an air dart gun at their balloon, once or several times, so that the balloon descends down as close as possible to its point of launch? How does this answer depend on where the initial dart strikes the balloon's surface, and on how soon after the first strike they respond?

In a simple approximation, a physicist's office could be described as a room with a desk in it. Because the physicist is really busy with teaching and submitting papers, they do not regularly remove the stacks of papers accumulating on the desk. This, unfortunately, implies that they cannot remove the dust on the desk. On the other hand, they hoover the floor of the office once a week. Making reasonable assumptions about the dust production, dust properties, airflow, geometry, etc, estimate the average amount of dust to be found on the desk in the steady state.

Optional suggestions:

• You can model the rate of dust deposition in the office as constant in time and space.

• The airflow may be approximated as highly chaotic and its moments as constant in time, thus the macroscopic motion of the dust particles induced by the airflow is equivalent to an effective temperature (you may assume the Brownian motion due to the actual temperature to be negligible).

• It is reasonable to assume a contact interaction between the dust particles and the floor as well as the desk surface (i.e. the chemical potentials are shifted by a constant term).

• You may neglect the effect of the papers on the desk.

• If you want to make the model even more realistic, you can add another interaction term proportional to the dust density on floor and desk surface, this models the short-range interaction between different dust particles (which you may assume to be negligible in the phase in which dust particles move freely in the air, due to their low density).

• You may assume the desk surface area to be small in comparison to the cross-section of the office whenever this approximation yields simplifications.

A culinarily challenged amplitudologist decided to bake some cookies. To his horror, he discovered that his baking sheet has spontaneously broken a Z₂ symmetry: a formerly flat metal sheet, it would bulge up and down, picking one of the two vacua under a slight application of pressure.

Why would a metal sheet bulge like that? Why was the symmetry broken? Devise a toy model for the baking sheet that would exhibit this symmetry-breaking property. Make whatever assumptions you find reasonable.

There exist vacuum geometries which are partially expanding (positive cosmological constant, c.c.) and partially collapsing (negative c.c.). Find such a geometry, and write it the following form:

ds² = −f(r)dt² + dr²/f(r) + R(r,t)² dΩ²

Where dΩ² are some additional degrees of freedom warped by R(r,t). Explain why you think your solution satisfies the criterion requested. Then, add in a spherical shell of massive particles with infinitesimally low energy density at the interface(s) of the expanding and collapsing regions of the spacetime. What criterion must you now impose on a generic positive-c.c. bubble of your universe?

Suppose you have a chain of N spins placed adjacent to one another around a circle. Some of these spins may be entangled within M subchains of indefinite length, where 1 ≤ M ≤ N. You are provided with the density matrix ρ corresponding to the system.

1. Construct an efficient algorithm which will isolate the M subchains of the system. Assume that these subchains will be visible in the form of ρ within a finite number b of bases for your Hilbert space. Include an analysis of the complexity of your algorithm. Do so assuming that you have crafted this spin chain in a laboratory so that entangled subchains will consist only of adjacent spins.

2. A competing scientist sees that you are close to a result on your spin chains breakthrough paper and, in a dastardly attempt to jeopardize your grant funding, decides to rearrange the spins in your apparatus. Thankfully, this evildoer does so without disrupting the entanglement of the spins. Modify your algorithm to relax the assumption that subchains consist of adjacent spins. You may now encounter troubling exponential runtimes, so try your best to reduce them.

Bacteria, such as E. coli, can collectively look for food via a chemotaxis signaling network. When bacteria eat food, they produce attractant molecules in the environment (a liquid medium). By sensing the gradient of attractant concentration, bacteria (in a run-and-tumble motion) swim as random-walkers with bias.

A system of PDEs – the Keller-Segel equations – is often used to describe bacteria chemotaxis:

∂ₜb − Dₑ∇²b + κ∇(b∇c) = αb

∂ₜc − Dᴄ∇²c = βbf

∂ₜf − Dᶠ∇²f = −γb

Here b stands for the bacterial density, c for the attractant concentration, and f for food concentration. The parameters Dₑ, Dᴄ and Dᶠ are the diffusivity coefficients of the bacteria, the chemoattractant molecules and the food molecules. The parameter κ is the sensitivity of bacteria to the chemo concentration gradient, α is the bacterial growth rate, β is the chemo production rate, and γ is the food consumption rate.

Consider the dynamics of a bacterial colony in an environment of volume V with a hollow shell in the center (volume Vₛ ≪ V). Let the hollow shell have a small hole (opening area Aₛ) on it to connect the inside and the outside. Assume that in the beginning the food concentration is the same everywhere, and we inoculate some bacteria into the environment. Your preliminary experiments show that the bacterial colony collapses into the hollow shell. Model this behavior theoretically and numerically. You may need to modify the Keller-Segel equations.

It is well-known that in a flat spacetime a finite temperature can lead to the screening of electric fields (in Debye mass phenomena, the field A₀ acquires a mass). It is also known that in Rindler spacetime there is a finite temperature due to quantum effects. Does this temperature lead to the emergence of Debye screening in Rindler space? If so, why?

Compute the mean effective resistance R̄ₐₑ of a statistical mechanical ensemble of random graphs of resistors connecting two points A and B in a two-dimensional plane. Assume that each vertex has exactly 4 edges of constant resistance R extending from it. When you are done, compute the mean effective spring constant assuming the resistors are now springs.

Consider particles of the same mass $m$, with a pairwise interaction given by a repulsive potential proportional to the inverse square of the distance between them:

1. For a two-body problem with initial positions and velocities given by

you are asked to find the particle trajectories.

2. Now, for a three-body problem with an additional particle

find the final velocities.

Consider a hot rotating spherical body in vacuum. It radiates electromagnetic waves which are expected to be polarized. Find the spatial dependence of the polarization far away from the radiating body. You may find it convenient to start with a cylindrically symmetric rotating thermal radiation.

A free massive scalar field $\varphi$ in 1 + 1 dimensions is described by

In many physical problems it is useful to promote coupling constants to dynamical fields. We can do the same with the mass $m$ considering the following partition function:

Study the stability of the following vacuum: $\varphi = 0$, $m = m_0 = \text{const} \neq 0$. Consider two cases: infinite volume and a finite interval $[0, L]$ in the $x$ direction with Dirichlet boundary conditions for $\varphi$ and free boundary conditions for $m$.

In this problem you are asked to calculate analytically or numerically the free energy as a function of temperature for the following three models:

1. The 2-dimensional Ising Model with an interaction

Where $\langle ij\rangle$ are adjacent vertexes.

2. The 3-dimensional Ising Model with an interaction

Where $\langle ij\rangle$ are adjacent vertexes.

3. The 3-dimensional tensor model with an interaction

Does this model experience phase transition? If yes, what is the order of this transition?

Peter The Experimentalist caught a finite number of lobsters in Maine. He decided first to play with them before having a nice dinner. Peter put these lobsters on an one-dimensional lattice. It is forbidden for lobsters to be at the same site of the lattice at the same moment. After that Peter waits N minutes, during a minute only one lobster can make only one hop to the left or to the right. Peter wants to know the possible final states with their multiplicities if the lobsters are allowed to hop only to the right {nᵢᴿ} or to the left {nᵢᴸ}. You are asked to prove or disprove the following statements:

1. Σnᵢᴸ = Σnᵢᴿ

2. Σ(nᵢᴸ)² = Σ(nᵢᴿ)²

3. If we are given a pattern of moves L or R and also the possible final states with multiplicities {nᵢᵃʳᵇ}, then Σ(nᵢᵃʳᵇ)² doesn't depend on the choices L or R but on the number of hops.

Even start out initially with quantum fuzziness, a macroscopic object will eventually becomes classical in a very short time due to the loss of quantum entanglement through interactions with the environment. Obtain this time scale, given the mass of the environment particles m (very small compare to that of the macroscopic object), the collision rate Γ between the object and the environment particles, the thermal equilibrium temperature T of the environment and the spatial resolution L.

A 2d metallic square K = [0, a] × [0, a] has a given resistivity density tensor:

ρ̂ = [[ρₓₓ, ρₓᵧ], [−ρₓᵧ, ρₓₓ]], E⃗ = ρ̂j⃗

There is a potential difference between two opposite sides of the square. In this problem you're asked to calculate the full current and the resistivity.

Can you engineer a dielectric permittivity tensor such that a real charge at some distance to an infinite half-space filled with a dielectric will generate an induced (real) point-like charge inside the medium? Note: a medium with such permittivity tensor might be not realizable, at least in a static (equilibrium) setup.

There is an elastic magnetized ferromagnetic rod in thermal equilibrium with a medium below the Curie temperature. The rod moves along the symmetry axis and approaches a wall. After the collision, the rod may start rotating around its symmetry axis. Assuming that the collision with the wall happens adiabatically, find the value of the angular momentum after the collision. How does it depend on material parameters and initial parameters of the motion? Does the angular momentum stay constant after the collision?

Let us investigate the effect of the Universe expansion on the matter. Study the hydrogen atom and for simplicity assume that the expansion is "turned on" at some moment. Treating the time-dependence as a perturbation, calculate the energy radiated by the atom.

There is a gravitating cloud of quantum particles of the same type with spin S. They do not interact with electromagnetic field - the charge and all multipole moments are zero. Suggest a method to obtain the spin of one particle if you know their number N, given you can measure and create any external gravitational field (obeying the Einstein equation). Express explicitly the spin value in terms of the quantities measured and describe your procedure.