- The project studies the paper [501]. Recall that the avalanche dynamics evolves on two time scales. The first, denoted by t, counts the numbers of energy units added to the pile.In between each
- We consider the outbreak of a transmittable disease in an infinite population. The disease is characterised by its ability η ∈ [0,1) to infect. Each individual has a specific level of resistance R
- Study numerically the fluctuations in the process in Eq. (13.1) with η(t) given by Eq. (13.2). Try, from the changing time dependence of the fluctuations (the variance and the autocorrelations), to
- Simulate the process in Eqs. (13.3) and (13.4) in the vicinity of the point at (X̅,0) and try to use as an early warning signal the logarithmic time dependence considered in Eq. (1) of
- Being able to forecast approaching transitions or tipping points in complex systems is of great importance and very difficult. Since no well-established consensus concerning the most viable approach
- Consider a population of N agents on a network. Let nb(i) denote the set of neighbours of agent i, i.e. if j ∈ nb(i) then j is a neighbour of i. The opinion xi of an agent can assume one of the
- The voter model, see e.g. [141], and the Axelrod model of cultural trait dynamics [31] have inspired very many agent-based studies of the dynamics of opinions and cultural evolution. This project
- The time between events in SOCmodels is typically exponentially distributed, indicating no correlations between events. This is often in contrast to observed behaviour. Solar flares is an example of
- Simulate the Manna model on a square lattice. The model is defined in the following way, see [276, 354]. Each site can contain zero or one particle. New particles are added to randomly chosen sites.
- Carefully go through the arguments leading to Eq. (12.23). Next derive Eq. (12.27).Equation (12.23)Equation (12.27) 1+1 0 1+1 z'dz = J P(t) =
- Consider the logistic mapDerive analytically an expression for the fixed point x* given by x* = ƒ(x*) and the point xtan for which ƒ(x) has a tangent parallel to the identity, i.e. ƒ(xtan) = 1.
- Machine learning, chaos and intermittency are discussed in [237]. Startwith the abstract and discussion sections and then skim read the paper with the aim of making a brief overview of how machine
- Derive the different expressions for the entropies S[ p] corresponding to the different functional forms of W(N) in Box 9.1.
- (a) Discuss what it can be that makes the Lempel–Ziv complexity of the EEG signal higher the more aware the brain.(b) Do we expect our brain activity to produce more new ‘thought patterns’ if
- Consider three stochastic variables Xt, Yt and Zt which assume the values −1 or 1 for t = 1,2,3, . . .. We assume that Z can swap freely between −1 and 1 with constant probability μZ at each
- Show that the normalisation constant A in Eq. (10.57) is given byEq. (10.57)Generalise the expression in Eq. (10.57) to the non-symmetric random walker for which p+ ≠ p− and derive expressions
- Let n(x,t) denote the density of random walkers at position x ∈ Z at time t ∈ Z. At each time step t a walker makes a move x → x + Δ, where Δ = + 1 or Δ = −1 with probability 1/2. (a)
- Consider a two-valued signal ƒ(t) ∈ {−A,A}. Assume that the probability that ƒ(t) switches during the time interval dt is constant and given by νdt. (a) Sow that the autocorrelation function
- Consider a discrete-time random walker moving on the set X = {na|n ∈ Z}, where a is the spacing between the discrete positions. Let p+, p0 and p− denote the probability that the walker from time
- Let n(x,t) be determined by the diffusion equationwhere g(x,t) is a given function.(a) Use Fourier transform to express n(x,t) in terms of the Fourier transform an(x,t) an(x,t) n(x, t) =y. Y + B- +
- Consider a population consisting of individuals that are either of opinion A or opinion B. Initially the entire population is of opinion A. New information arrives, which is scrutinised by everyone
- Consider the merger process described in [472]. The process is controlled by the merger probabilitywhere α = 3/2 is considered in the paper.(a) Simulate the process for α = 3/2, say α = 1 and α =
- Simulation of a population of N agents. Each agent i is allocated a willingness parameter wi ∈ [0,1] and possesses one of two opinions A or B. Focus on two choices for the distribution of wi. f(p).
- This project is of direct relevance to the concepts of a giant component in Sec. 8.4.The process consisting of percolation on a lattice is another paradigmatic model, which conceptually is of great
- Consider a network given by the adjacency matrix aij = aji = 1 with probability p and aij = aji = 0 with probability 1 − p.(a) In the limit of large N and fixed pN, compute the degree distribution
- Think of throwing a coin. Let X = 1 when the outcome is heads and X = −1 when the outcome is tails. We assume the coin is biased and that PX(1) = p and PX(−1) = 1−p. Compute the entropy of X
- An example of dependent variables with zero correlation coefficient. We consider thefollowing two cases. Let X be uniformly distributed(I) on the set {0,1,2, . . . ,N};(II) on the set {−N, − N +
- Consider two Ising spins in the canonical ensemble with a Hamiltonian given by H = −S1S2. Compute the correlation coefficient CX1X2 and the mutual information I(X1;X2) between the two variables X1
- Derive the expressions for the mutual information given in Eq. (9.20).Equation (9.20) I(X; Y)H(X)+ H(Y)- H(X, Y) = H(X, Y) - H(XY) - H(YX) =H(X) H(XY) - = H(Y) H(YX).
- Use the relation between the transfer entropy and mutual information to derive Eq. (9.27).Equation (9.27) = DTEX-X, TEXX-TEX(-)X)-
- We generalise the set of Eqs. (9.14) to include an additional dependence between X and Z:Equation 9.14 x(t) = $1(t) y(t)=ax(t z(t) =By(t 1) + 2(1) 1)+3(1).
- Show that in the limit q → 1 the Tsallis entropy in Eq. (9.36) becomes equal to the entropy in Eq. (9.31). And show that in the same limit the q-exponential in Eq. (9.38) becomes an ordinary
- Consider a general branching process. Let g(s) denote the generator function for the branching probabilities and let gZn (s) denote the generator function for the size Zn of the nth generation.
- We had stated in this chapter that the general Rényi entropies do not satisfy subadditivity; only the von Neumann entropy does. In Example 12.2, we constructed an explicit counterexample for the
- The von Neumann entropy satisfies a further inequality among three systems called strong subadditivity. \({ }^{20}\) As with subadditivity, strong subadditivity is only a property of the von Neumann
- In our analysis of hydrogen in Chap. 9, we discussed the period of recombination, the time in the history of the universe at which protons and electrons became bound and formed neutral hydrogen. We
- We had established an intriguing relationship between the path integral of the previous chapter and the partition function here through "complexification" of the time coordinate. In this problem, we
- We had mentioned that from the partition function \(Z\), all possible thermodynamical quantities can be determined. In this problem, we will use the harmonic oscillator's partition function of Eq.
- An extremely intriguing feature of quantum entanglement is a property that has been called the "monogamy of entanglement": a particle can only be maximally entangled with precisely one other
- In Example 12.1, we introduced the Hong-Ou-Mandel interferometer and presented an analysis of thinking about the photons produced by the laser as classical electromagnetic waves. In this exercise, we
- Let's consider the harmonic oscillator immersed in a heat bath of temperature \(T\). For any Hermitian operator \(\hat{A}\), we can define its thermal average denoted as
- In Sec.12.4.3, we motivated and introduced Bell's inequalities for unambiguous observation of quantum entanglement. There we demonstrated that a classical, random density matrix of two spins
- The quaternions are three quantities i, j, and k such that, along with the real number 1, they form the basis of a four-dimensional space over the real numbers. The objects i, j, and k have
- The Hong–Ou–Mandel interferometer is an experiment in which two identical photons are emitted from a laser toward separate mirrors that then reflect onto a beam splitter and can subsequently be
- Let’s see if we can make some more sense of the statement of subadditivity, and in particular, demonstrate that some particular Rényi entropies violate it.In this example, we will explicitly
- Let’s go back to our spin-1/2 friends and construct the density matrix for a collection of spin-1/2 particles. Let’s say that we have a partially polarized beam for which a fraction of 3/4 are
- As we explicitly calculated the path integral of the harmonic oscillator in the previous chapter, it is useful to calculate the partition function for the harmonic oscillator to compare the explicit
- In the next section, we will explicitly calculate the path integral for the harmonic oscillator. This will be highly non-trivial, and its calculation will involve many moving parts. Here, however, we
- After calculating the path integral for the harmonic oscillator, you might need to take a breath or two. To relax, in this example, we will calculate the path integral for the free particle, using
- The appearance of the classical Lagrangian and action in the path integral might be surprising given how we had constructed quantum mechanics throughout this book, but it was actually hidden in plain
- In the evaluation of the path integral for the harmonic oscillator, we had left the classical action unevaluated. It's high time to fix that.(a) From the Lagrangian for the harmonic
- In evaluating the path integral of the harmonic oscillator, we had to perform a Gaussian integral for which the factor in the exponent was a Hermitian matrix \(\mathbb{A}\) sandwiched between an
- As emphasized in this chapter, calculating the path integral for even very simple quantum systems is highly non-trivial. In Example 11.2, we calculated the freeparticle path integral one way, and
- In this chapter, we only considered path integrals for one-dimensional systems, but in some cases it is straightforward to calculate the path integral for a system in three spatial dimensions. As an
- In the calculation of the path integral, we of course have to sum over all possible paths between the initial and final positions. However, it can be useful to consider what contributions to the path
- Consider the familiar double-slit experiment: a quantum mechanical particle of mass \(m\) starts at some position to the left of the slits at time \(t=0\). As time evolves, the wavefunction of the
- While we have focused on the Lagrangian and action for point particles in this chapter, it is possible to formulate the Schödinger equation itself through the principle of least action with a
- Let’s see how this formalism works in an explicit example. Let’s consider a potential that is the infinite square well that has a little bump in the center, as shown in Fig. 10.1. That is, the
- Let’s test this out for the infinite square well, where we will set the width a = 1 of the well for simplicity. That is, the potential we consider isWe know what the ground-state energy is in this
- While this seems pretty magical, the biggest challenge will be inverting the Hamiltonian in the first place. As we saw with the variational method, an example can be helpful in demonstrating the
- The WKB approximation and the Bohr–Sommerfeld quantization condition can be used to calculate the energy levels of the hydrogen atom, as well, and was actually the system in which it was initially
- While the Bohr–Sommerfeld condition sometimes gets the energy eigenvalues exactly correct, it can also be used for systems where the exact solution is not known. In this example, we will estimate
- It's useful to see how our quantum perturbation theory works in a case that we can solve exactly. Let's consider a two-state system in which the Hamiltonian
- Let's see how the variational method works in another application. Let's assume we didn't know the ground-state energy of the quantum harmonic oscillator and use the variational method to determine
- While we introduced the variational method and the power method both as a way to approximate the ground state of some system, they both can be used to approximate excited states as well, with
- Let's now study the power method for estimating the ground-state energy, applied to the quantum harmonic oscillator. For this problem, we will work with the
- The anharmonic oscillator is the quantum system that is a modification to the harmonic oscillator, including a term quartic in position:\[\begin{equation*}\hat{H}=\frac{\hat{p}^{2}}{2 m}+\frac{m
- We had introduced quantum mechanical perturbation theory as analogous to the Taylor expansion. As such, one might expect that as higher and higher orders are calculated, a better approximation of the
- The WKB approximation and the Bohr-Sommerfeld quantization condition worked perfectly for calculating the energy eigenvalues of the infinite square well and the hydrogen atom. Does it work for the
- We can quantitatively study this claim, that for the hydrogen atom, we do not need to invert the Hamiltonian to use the power method to estimate its groundstate energy. In this exercise, we'll just
- In our introduction of the hydrogen atom, we assumed that the electron orbited the proton with a speed significantly smaller than the speed of light, \(c\). However, we can include the effect of a
- We have evaluated the ground-state wavefunction of hydrogen in Eq. (9.48), and in this example, we will construct the first excited state n = 2, ℓ,m = 0 wavefunction. This can then be used, along
- The Zeeman effect is the phenomenon of splitting energy levels with different angular momentum due to the presence of a weak external magnetic field, named after Pieter Zeeman who first observed
- Our analysis of the hydrogen atom simply extends to any element which has been ionized to have a single electron orbiting the nucleus. In this problem, we will consider such an atom, whose nucleus
- In introducing the Laplace-Runge-Lenz vector and the canonically quantized hydrogen atom, there were a number of Poisson brackets for which we just stated the result, without explicit calculation.
- The virial theorem is a statement about the relationship between the kinetic and potential energies for a system bound in a potential that is a power of relative distance between particles. For such
- The form of the potential of the hydrogen atom is of course special, because it originates from Coulomb's law of electric force through canonical quantization. However, we can imagine a general form
- We explicitly constructed the ground-state and first excited-state wavefunctions for the hydrogen atom in this chapter, but we can go further. The LaplaceRunge-Lenz operator commutes with the
- The visible light spectrum ranges from wavelengths of \(380 \mathrm{~nm}\) (violet) to \(750 \mathrm{~nm}\) (red). If a hydrogen atom were excited, what color light would it emit and therefore appear
- In Example 9.2, we introduced the Zeeman effect as a splitting of the energy levels of states in hydrogen due to the presence of a weak, external magnetic field. In that example, we just studied the
- To determine the eigenvalues of the Hamiltonian of hydrogen we constructed the operators \(\hat{T}\) and \(\hat{S}\) from angular momentum and the Laplace-Runge-Lenz operator. The actual form of
- In this chapter, we studied the Lie algebra of rotations in great detail, but didn't construct the Lie group through exponentiation, except in some limited examples. In this problem, we'll study the
- The Jacobi identity is a requirement of the commutation relations of a Lie algebra that ensures the corresponding Lie group is associative. For elements \(\hat{A}, \hat{B}, \hat{C}\) of a Lie
- The Killing form of a Lie algebra provides the definition of normalization of operators in a particular representation of the Lie algebra. For representation \(R\) of \(\mathfrak{s u}(2)\), the
- We've studied coherent states in the context of the harmonic oscillator and the free particle, and our formulation of angular momentum suggests that there is a definition of coherent state in that
- In Example 8.3, we introduced a simple quantum system involving an electrically charged, spin- \(1 / 2\) particle immersed in a uniform magnetic field. In this exercise, we will continue the study
- A very powerful technique for expressing amplitudes of scattering processes in quantum field theory, the harmonious marriage of special relativity and quantum mechanics, is through spinor helicity,
- The \(\mathfrak{s u}(2)\) algebra consists of three operators in the basis of its Lie algebra which are not mutually commutative. We could also consider a Lie algebra for which the basis is a single
- We had illustrated that rotations in three dimensions are non-commutative or non-Abelian through the example of rotating a coffee cup about two orthogonal axes in two different orders. While a very
- 7.2 For this problem, you will use properties of the reflection \(A_{R}\) and transmission \(A_{T}\) amplitudes that we derived in this chapter:\[\begin{align*}&A_{R}=\frac{m V_{0}}{k^{2}-m
- 7.3 From the construction of the S-matrix, we had identified the interaction matrix as that which encodes the non-trivial reflection and lack of transparency off a potential. In this problem, we'll
- 7.4 In this chapter, we considered the scattering of momentum eigenstates off localized potentials. Of course, momentum eigenstates are not states in the Hilbert space, and so we have to consider
- 7.5 We introduced an operator \(\hat{U}_{p}(x, x+\Delta x)\) in this chapter that represented generalized translations for which momentum was not assumed to be conserved. We used that operator to
- 7.6 We have stressed that the wavefunction of a quantum mechanical particle does not directly represent the trajectory of that particle through space. This is very unlike the way in which we
- 7.7 We demonstrated that the S-matrix is unitary and that its poles correspond to \(L^{2}\)-normalizable eigenstates of the Hamiltonian, at least in the case of the narrow step potential. Assuming
- 7.8 Consider the potential illustrated in Fig. 7.19, where there is a finite potential barrier of height \(V_{0}\) and width \(a\), that terminates at \(x=0\) at which the potential becomes infinite
- 7.1 Let's first consider a free particle, whose wavefunction can be expressed as\begin{equation*}\psi(x, t) = \int_{-\infty}^{\infty} \frac{d p}{\sqrt{2 \pi \hbar}} g(p) e^{-i \frac{E_{p} t - p
- At its core, the Hamiltonian of the harmonic oscillator involves the operator that is the product of the raising and lowering operators, ˆ a† ˆa. We had shown that the physically sensible
- With the expression for the ground-state wavefunction, let’s keep going and construct the wavefunction of the first excited state, ψ1(x), of the harmonic oscillator.Recall that the relationship
- With this abstract form of a coherent state in Eq. (6.103), defined through the action of the raising operator on the ground state of the harmonic oscillator, let’s see if we can express it in a

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