Theoretical Models. → ∞ We present a brief overview of the theoretical models studied in this thesis. One of the main models we shall use in this thesis to study the dynamics of relaxation and thermalization in an isolated quantum system is xxx Xxxxxxx Ye Kitaev model (SYK) [39, 40]. This is a strongly interacting quantum mechanical system of N Majorana fermions with all-to-all random couplings. which is exactly solvable in the IR limit when N . The strongly coupled IR fixed point has an emergent reparametrization symmetry where the effective dynamic is described by the Schwarzian action. The SYK model can be argued to be the ideal (solvable) representative of many-body quantum systems. It displays maximally chaotic behavior, as defined in Section 1.3.1, with an exponentially growing OTOC and a Lyapunov coefficient that, in the deep IR, saturates the bound (1.14), but at its heart it also obeys ETH [41]. These properties are precisely what makes the SYK an interesting playground for studying quantum chaos, quantum gravity and information scrambling. Moreover, as we will briefly describe for completeness in Section 1.5, these properties are also typical for 2D dilaton gravities which describe the near horizon physics of extremal black hole [42, 43, 44]. And a version of SYK built on charged Dirac fermions instead of Majoranas [45, 46, 47] has proved to be a useful toy model for studying strange metals and high- temperature superconductivity [19, 48]. The other model we shall study is the Transverse Field Ising model (TI), which is a set of Pauli matrices on a 1D chain, a well-known model in the study of quantum thermalization [49, 50]. On one hand, it is also solvable: using xxx Xxxxxx-Xxxxxx (1.46) and Xxxxxxxxxx (1.51) transformations can be shown it is equivalent to free fermions (1.52) (Section 1.4.2). On the other hand, it is representative of systems with a second-order quantum phase transition. The quantum phase transition occurs for an infinite chain at zero temperature T = 0 when the transverse magnetic field equals the hopping strength hx = J. At this quantum critical point, the system becomes strongly entangled and it has distinct dynamics even at finite temperatures and finite system sizes. The last considered model is a generalization of the TI known as the Mixed Field Ising (MFI) Section (1.4.3), a fruitful playground for, among others, studying quantum thermalization, quantum chaos and information scrambling [51, 52, 53, 54]. Its usefulness comes from the abilit...
Theoretical Models. Forensic mental health services place significant emphasis on assessing and treating aggression in their day-to-day practice. The Risk-Need-Responsivity (RNR) model (Xxxxxxx et al., 1990) has been one of the most influential theoretical models for assessing and managing risk of aggression in forensic mental health services and consists of three principles (Xxxxxxx & Xxxxx, 2010). The risk principle proposes that treatment efforts should be matched to service users’ level of risk (i.e. service users at the highest risk of aggression should receive the most intense level of treatment). The need principle states that characteristics of the service user which are: i) related to their risk of aggression and ii) changeable (dynamic risk factors, see section below) should be the primary focus of these treatment efforts. The responsivity principle highlights the need for empirically supported treatment models which should be tailored to service users according to their motivation, strengths, and abilities. The need principle is the core of the RNR model as it dictates which factors associated with recidivism should be targeted through treatment efforts (Xxxx & Xxxxxx, 2016). The Good Lives Model (GLM) is another influential model of service user rehabilitation in the context of aggressive behaviour and shares some of features of RNR. It is a strength-based approach to rehabilitation, focussing on both the risk factors of an individual and developing their strengths (Laws & Xxxx, 2011). Aggressive behaviour represents flawed attempts to secure important goals in life, with dynamic risk factors serving as barriers to these goals (Xxxx et al., 2011). Identifying and modifying these dynamic risk factors is therefore an important component of treatment.
