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Having a support [MATH] , it remains to present an appropriate signature [MATH] allowed by [MATH] Instead of representing the signature for each support and each [MATH] explicitly we rather use again a general scheme that works for almost all ESSP atoms. More exactly, given a set [MATH] defined by a certain row of the ... |
[EQUATION] Note that, by [MATH] and [MATH] for all [MATH] such a region actually inhibits all events of [MATH] at all states of [MATH] As already mentioned, some ESSP atoms [MATH] requires a special treatment and need to be discussed individually. However, these cases are very seldom and they will be discussed at the a... |
If the ESSP for [MATH] is proven then it remains to argue for the SSP. This will be done at the very end of this section in Lemma 8.18 |
Lemma 8.1 The key event is inhibitable. Proof 8.2 An input key region inhibits [MATH] already in [MATH] except for the states [MATH] and at the relevant states of [MATH] and [MATH] Therefore, it only remains to show that [MATH] is inhibitable at the relevant states of [MATH] and at [MATH] The first row of the next tabl... |
Lemma 8.3 The events [MATH] are inhibitable. Proof 8.4 Let [MATH] and [MATH] such that [MATH] The regions of the first two rows prove [MATH] to be inhibitable in the TSs it occurs in, the last row is dedicated to the states of the other TSs. |
Lemma 8.5 The events [MATH] are inhibitable. Proof 8.6 The first row of the following table is dedicated to the inhibition of [MATH] at certain states of [MATH] and the sources/sinks of [MATH] in [MATH] [MATH] and [MATH] Hence, for [MATH] and each generator [MATH] installed by [MATH] we, firstly, assume the sinks [MATH... |
Lemma 8.7 The events [MATH] are inhibitable. Proof 8.8 The first row of the following table is, firstly, dedicated to the inhibition of [MATH] at [MATH] and [MATH] To do so, for [MATH] and all generators [MATH] installed by [MATH] we assume the sinks (sources) of [MATH] -labeled transitions to be included (excluded) an... |
Lemma 8.9 The events [MATH] and [MATH] and [MATH] are inhibitable. Proof 8.10 If [MATH] [MATH] and [MATH] such that [MATH] then the first row of the following table, firstly, proves [MATH] and [MATH] at [MATH] , that is, [MATH] for all [MATH] to be inhibitable at [MATH] Moreover, these region inhibits the event [MATH] ... |
The second row, is dedicated to the inhibition of [MATH] at the states [MATH] Here, if [MATH] we assume [MATH] to be included and if [MATH] we include [MATH] For simplicity, we refrain from presenting these states explicitly. |
The third row, inhibits the events [MATH] [MATH] ) for [MATH] [MATH] ) at the remaining states of [MATH] Lemma 8.11 The event [MATH] is inhibitable. |
Proof 8.12 The following table proves [MATH] to be inhibitable in [MATH] Lemma 8.13 The events of [MATH] and [MATH] are inhibitable. |
Proof 8.14 We proof the lemma by presenting regions for an arbitrary [MATH] that inhibits the events [MATH] at all relevant states of [MATH] Let [MATH] and [MATH] The first row is dedicated to the inhibition of [MATH] at [MATH] and the second row proves both of them to be inhibitable at [MATH] , where [MATH] are the st... |
(1) [MATH] (2) [MATH] (3) [MATH] (4) [MATH] (5) [MATH] (6) [MATH] Note that, for the third row, we rather have need of some further states of [MATH] because of some incoming/outgoing c-labeled transitions in [MATH] Depending on [MATH] the actual presentation of these states would render a lot of case analyses as, for e... |
Lemma 8.15 The events [MATH] and [MATH] are inhibitable. Proof 8.16 For arbitrary [MATH] we present regions of [MATH] that inhibits the event [MATH] and [MATH] at all states of [MATH] Having this, by the arbitrariness of [MATH] and [MATH] this implies that [MATH] are inhibitable in [MATH] |
Let [MATH] arbitrary but fixed. Let [MATH] such that [MATH] and, for [MATH] , let [MATH] [MATH] ) if [MATH] [MATH] ). Finally, let [MATH] and [MATH] for [MATH] For [MATH] , we get: |
(1) The states [MATH] are the sources of [MATH] and the sinks of [MATH] in [MATH] (2) The states [MATH] are the sinks of [MATH] and the sources of [MATH] in [MATH] |
We now define subsets of [MATH] which will be used to combine supports of regions of [MATH] (1) [MATH] (2) [MATH] (3) [MATH] (4) |
[MATH] (5) [MATH] (6) [MATH] (7) [MATH] The following table proves [MATH] to be inhibitable at all states in question besides of [MATH] and [MATH] Observe that, if [MATH] then [MATH] |
To prove that [MATH] is inhibitable at the remaining states [MATH] and [MATH] , too, we need to additionally involve the variable event [MATH] , which, by the further occurrences of [MATH] in other translators, renders a lot of case analysis. However, going through all the cases is tedious and renders no considerable i... |
(1) For [MATH] we put exactly the states [MATH] into the support which makes the arcs labeled with [MATH] and [MATH] incoming and [MATH] outgoing. |
(2) If [MATH] such that [MATH] and [MATH] then if for [MATH] and [MATH] the state [MATH] is a source of [MATH] in [MATH] then put the states [MATH] into the support. For [MATH] [MATH] ) this makes [MATH] [MATH] ) incoming events. |
(3) If [MATH] such that [MATH] and [MATH] then if for [MATH] and [MATH] the state [MATH] is a sink of [MATH] in [MATH] then put the states [MATH] into the support. For [MATH] [MATH] ) this makes [MATH] [MATH] ) incoming events. |
(4) If [MATH] such that [MATH] and [MATH] then choose the [MATH] -part of the support corresponding to as which of [MATH] the events [MATH] and [MATH] occur, cf. Figure This condemns at most some additional [MATH] -events or [MATH] -events to be incoming events. |
(5) Choose, in dependence of the border crossing [MATH] -, [MATH] - or [MATH] - events the necessary additional states of [MATH] If [MATH] is incoming then choose [MATH] , making [MATH] exiting, and [MATH] and [MATH] if [MATH] to be included. If [MATH] is incoming then include [MATH] , making possibly [MATH] entering a... |
The given construction plan yields a region that inhibits [MATH] at the remaining states [MATH] and [MATH] of [MATH] Altogether, this proves [MATH] to be inhibitable in [MATH] Moreover, by the symmetry of the occurrences of [MATH] and [MATH] the event [MATH] is inhibitable at all states of [MATH] in question besides of... |
(1) [MATH] (2) [MATH] (3) [MATH] (4) [MATH] (5) [MATH] (6) [MATH] (7) [MATH] (8) [MATH] (9) [MATH] (10) [MATH] (11) [MATH] Now, to enrich the set [MATH] to a fitting support of [MATH] if [MATH] , respectively the set [MATH] if [MATH] , we have to take the remaining generators into account. That is, for all the other ge... |
To show that the inhibition of the key event at the key state in [MATH] implies the ESSP, it remains to show that the unique events are inhibitable, too. Each unique event occurs exactly once in [MATH] and, therefore, only in one single TS. Regarding this fact and the symmetry of the gadget TSs, mainly the generators, ... |
Lemma 8.17 (Without proof) The unique events are inhibitable. Finally, the next lemma states that if [MATH] is inhibitable at [MATH] in [MATH] then [MATH] has the SSP. |
Lemma 8.18 If [MATH] is inhibitable at [MATH] in [MATH] then [MATH] has the SSP. Proof 8.19 For simplification, in the following by a key region we mean a region of [MATH] that inhibits [MATH] at [MATH] Firstly, if [MATH] and [MATH] in [MATH] such that [MATH] is inhibitable at [MATH] by an input region, that is, there ... |
8.2. Concluding the ESSP and the SSP for [MATH] Let [MATH] and [MATH] and [MATH] In this section we present explicitly supports showing that the inhibition of the key event at the key state implies the [MATH] -(E)SSP for [MATH] The following observation helps us to reduce the number of ESSP atoms that have to be solved... |
[EQUATION] Note, that this approach to define a signature implies that the signature only depends on the given support [MATH] and the switch [MATH] Therefore, for the sake of simplicity, in the sequel we often refer to a given support [MATH] as to the region [MATH] which it allows and, e.g., say [MATH] inhibits [MATH] ... |
In the following proofs, for [MATH] if we have given one of [MATH] or [MATH] as an element of [MATH] then the value of the others is assumed to be determined by the following definitions: |
(1) given [MATH] [MATH] and [MATH] (2) given [MATH] [MATH] and [MATH] (3) given [MATH] [MATH] and [MATH] Lemma 8.20 The key event is inhibitable. |
Proof 8.21 In the following for [MATH] let [MATH] and [MATH] arbitrary but fixed and [MATH] such that [MATH] For [MATH] let [MATH] [MATH] [MATH] ) if [MATH] [MATH] ). |
If [MATH] [MATH] ) then a key region inhibits [MATH] at all states of the key union besides [MATH] and [MATH] and [MATH] Furthermore, with respect to [MATH] a key union inhibits [MATH] at all states of [MATH] and at [MATH] [MATH] ). |
Hence, by the arbitrariness of [MATH] and [MATH] and the symmetry of the translators, to prove [MATH] to be inhibitable in [MATH] it is sufficient to present regions that show the inhibition of [MATH] at [MATH] and at [MATH] [MATH] ) and at [MATH] [MATH] ) and at [MATH] To attack these challenge we define the following... |
(1) [MATH] [MATH] (2) [MATH] (3) [MATH] (4) [MATH] (5) [MATH] and [MATH] (6) [MATH] and [MATH] (7) [MATH] and [MATH] (8) With the index set [MATH] corresponding to the affected a-events we have |
(a) [MATH] (b) [MATH] (c) [MATH] (d) [MATH] (9) With the index set [MATH] corresponding to the affected w-events we have (a) [MATH] |
(b) [MATH] (c) [MATH] (d) [MATH] (10) [MATH] (11) [MATH] [MATH] (12) [MATH] (13) [MATH] and [MATH] Using the just defined sets, the following table shows the inhibition of [MATH] at the states in question: |
Lemma 8.22 The events [MATH] and [MATH] are inhibitable. Proof 8.23 If [MATH] [MATH] ), [MATH] and [MATH] , then the lemma is justified for [MATH] if we show the inhibition of [MATH] [MATH] ) in [MATH] [MATH] ) and [MATH] [MATH] ). For [MATH] [MATH] ) the event [MATH] [MATH] ) is, besides of [MATH] [MATH] ), already in... |
(1) [MATH] (2) [MATH] (3) [MATH] (4) [MATH] (5) [MATH] and [MATH] (6) [MATH] and [MATH] For [MATH] [MATH] ) the set [MATH] [MATH] ) is a support that allows a signature such that [MATH] [MATH] ) is inhibited at [MATH] [MATH] ) and [MATH] [MATH] ). Finally, for [MATH] [MATH] ) the support [MATH] [MATH] ) can be used for... |
We now argue that [MATH] is for [MATH] and [MATH] inhibitable in [MATH] and [MATH] Firstly, we observe that for [MATH] [MATH] ) the key region inhibits [MATH] at [MATH] [MATH] ). Secondly, the region [MATH] of Lemma 8.20 inhibits [MATH] at all remaining states of [MATH] [MATH] ) besides of [MATH] The inhibition at [MAT... |
(1) [MATH] (2) [MATH] Then for [MATH] , respectively for [MATH] , the support [MATH] , respectively [MATH] , allows a signature such that [MATH] is inhibited at [MATH] That is, the inhibition of [MATH] in [MATH] is completed. We now argue for the states in question of [MATH] The inhibition at [MATH] is done for [MATH] ... |
Lemma 8.24 The event [MATH] is inhibitable Proof 8.25 For [MATH] [MATH] ) the support [MATH] [MATH] ),where (1) [MATH] (2) [MATH] |
(3) [MATH] allows an inhibiting region of [MATH] [MATH] ), Lemma 8.26 The events [MATH] and [MATH] are inhibitable. Proof 8.27 Let [MATH] and [MATH] be arbitrary but fixed. We need the following sets: |
(1) [MATH] (2) [MATH] , where (a) [MATH] (b) [MATH] (c) [MATH] (3) [MATH] (4) [MATH] , where (a) [MATH] (b) [MATH] In the sequel for [MATH] let [MATH] Let [MATH] and [MATH] and [MATH] The inhibition of [MATH] and [MATH] at [MATH] and [MATH] is allowed by the support [MATH] The support [MATH] allows the inhibition of [M... |
Lemma 8.28 The variable events [MATH] and [MATH] are inhibitable. Proof 8.29 Let [MATH] [MATH] [MATH] and [MATH] such that [MATH] For [MATH] let [MATH] [MATH] [MATH] ) if [MATH] [MATH] [MATH] ). Using the following sets: |
(1) [MATH] (2) [MATH] we have that [MATH] [MATH] ) is a support that allows the inhibition of [MATH] in [MATH] [MATH] ) and, therefore, in [MATH] [MATH] ). Similarly, we obtain the inhibition of [MATH] in [MATH] Finally, the set [MATH] [MATH] ) can be enhanced to a support of [MATH] [MATH] ) allowing the inhibition of ... |
Lemma 8.30 The event [MATH] is inhibitable. Proof 8.31 For [MATH] the event [MATH] occurs only in [MATH] Using the sets (1) [MATH] |
(2) [MATH] (3) [MATH] we have [MATH] , respectively [MATH] , as supports of [MATH] that, altogether, allow the inhibition of [MATH] at all states of [MATH] in question, besides of [MATH] Moreover, [MATH] is the only event that [MATH] and [MATH] require to be border crossing that occurs in TSs of [MATH] Hence, [MATH] an... |
[EQUATION] respectively [MATH] , is a support of [MATH] , respectively [MATH] , inhibiting [MATH] at [MATH] Lemma 8.32 The event [MATH] is inhibitable. |
Proof 8.33 The needed sets are: (1) [MATH] (2) [MATH] (3) [MATH] (4) [MATH] (5) [MATH] (6) [MATH] (7) [MATH] For [MATH] [MATH] ) the support [MATH] [MATH] ) allows the inhibition of [MATH] at [MATH] The same does [MATH] [MATH] ) for [MATH] The remaining states are [MATH] For [MATH] the set [MATH] is a support of [MATH]... |
Lemma 8.34 The event [MATH] is inhibitable. Proof 8.35 We need (1) [MATH] (2) [MATH] (3) [MATH] (4) [MATH] The set [MATH] is a support of [MATH] that allows the inhibition of [MATH] at all states in question besides of [MATH] The set [MATH] , respectively [MATH] , is a support of [MATH] , firstly, allowing the inhibiti... |
Lemma 8.36 The events [MATH] and [MATH] are inhibitable. Proof 8.37 The events [MATH] are inhibitable at [MATH] for [MATH] and [MATH] and [MATH] Hence, initially, we focus on the inhibition of [MATH] at the states [MATH] for [MATH] and [MATH] and [MATH] The following sets are needed: |
(1) [MATH] (2) [MATH] (3) [MATH] (4) [MATH] (5) [MATH] For [MATH] the set [MATH] is a support that allows the inhibition of [MATH] at [MATH] The set [MATH] , respectively [MATH] , is the same for the inhibition at [MATH] , respectively [MATH] By the symmetry of the occurrences of [MATH] in [MATH] , the event [MATH] can... |
Lemma 8.38 The events [MATH] are inhibitable. Proof 8.39 With (1) [MATH] (2) [MATH] (3) [MATH] (4) [MATH] (5) [MATH] the sets [MATH] and [MATH] are supports of [MATH] that, altogether, allow the inhibition of [MATH] in [MATH] for [MATH] and [MATH] Moreover, for the corresponding regions the event [MATH] is the only bor... |
Firstly, all [MATH] are inhibitable at [MATH] and [MATH] Secondly, we show [MATH] to be inhibitable at all further states of [MATH] in question, besides of [MATH] and [MATH] We do so by presenting respective supports for the switch [MATH] and the simple set complementation yields corresponding regions for [MATH] Finall... |
(1) [MATH] (2) [MATH] (3) [MATH] (4) [MATH] (5) [MATH] (6) [MATH] (7) [MATH] (8) If [MATH] then [MATH] For [MATH] the set [MATH] is a support that allows the inhibition of [MATH] at the remaining states of [MATH] , besides of the relevant states of [MATH] and [MATH] By set complementation we obtain corresponding suppor... |
We can now argue as follows: The region allowed by [MATH] inhibits [MATH] at [MATH] Moreover, the support [MATH] [MATH] ) allows a region that inhibits [MATH] [MATH] [MATH] ) at [MATH] [MATH] ). Finally, the region allowed by [MATH] inhibits [MATH] at [MATH] Hence, by the arbitrariness of [MATH] and [MATH] and the symm... |
To show that the inhibition of the key event at the key state in [MATH] implies the ESSP, it remains to show that the unique events are inhibitable, too. As each of these events [MATH] occurs exactly twice in [MATH] , that is, at a single backward and forward edge [MATH] , and, therefore, only in one single gadget TS, ... |
Lemma 8.40 (Without proof) The unique events are inhibitable. Finally, it remains to prove that the inhibition of [MATH] at [MATH] implies the SSP for [MATH] |
Lemma 8.41 For [MATH] the union [MATH] has the SSP. Proof 8.42 Firstly, the initial state of any TS [MATH] , the state with an incoming/outgoing unique event labeled transition, installed by [MATH] is separable from all the other states of [MATH] Secondly, if [MATH] then [MATH] is separable. Finally, for the solutions ... |
TS Separating Regions [MATH] Lemma 8.36 , Lemma 8.38 [MATH] Lemma 8.32 , Lemma 8.36 [MATH] Lemma 8.32 , Lemma 8.36 [MATH] Lemma 8.20 , Lemma 8.30 , Lemma 8.32 |
[MATH] Lemma 8.20 , Lemma 8.30 , Lemma 8.32 [MATH] Lemma 8.24 , Lemma 8.30 , Lemma 8.32 [MATH] Lemma 8.28 , Lemma 8.34 , Lemma 8.32 |
[MATH] Lemma 8.22 , Lemma 8.24 , Lemma 8.32 , Lemma 8.34 , Lemma 8.38 [MATH] Lemma 8.20 , Lemma 8.24 , Lemma 8.32 , Lemma 8.38 [MATH] |
# Source: arxiv 1806.03758 # Title: On critical dynamics and thermodynamic efficiency of urban transformations # Sections: all # Downloaded: 2026-03-03T04:47:18.444277+00:00 |
On critical dynamics and thermodynamic efficiency of urban transformations Abstract Urban transformations within large and growing metropolitan areas often generate critical dynamics affecting social interactions, transport connectivity and income flow distribution. We develop a statistical-mechanical model of urban tr... |
Keywords: Urban modelling, Thermodynamic efficiency, Maximum entropy principle, Phase transitions, Criticality, Fisher information |
Introduction A city is quintessentially a complex system consisting of multiple interacting agents such as individual residents, employment centres and transport infrastructure |
The complexity manifests itself through diverse spatial organisations: monocentric cities where most of the economic activity takes place at the CBD |
, polycentric cities with multiple subcentres (or “edge” cities) and dispersed sprawl (or “edgeless”) cities Moreover, cities can undergo transitions in their urban structures. Driving such transitions are changes in the factors determining economies and diseconomies of agglomeration for both firms and residents |
While urban dynamics have been extensively studied in the past, a unified framework centered on thermodynamics of urban transformations has not been yet developed (see a review |
). In particular, the analysis and modelling of urban transformations as phase transitions, defined in a rigorous thermodynamic setting, remains an open challenge, despite recent attempts in spatial economics over short time scales |
This paper aims to refocus the research field on Urban Thermodynamics, considering critical phenomena including phase transitions in a principled way, based on the underlying thermodynamic concepts (energy potentials, entropy, order parameters, etc.), for both equilibrium and nonequilibrium scenarios. This approach wil... |
We develop a statistical-mechanical model displaying phase transitions, using the maximum entropy principle in a dynamic setting, and define the thermodynamic efficiency of urban transformations. The model is calibrated to Greater Sydney Census data and is shown to exhibit a phase transition between a monocentric dispe... |
, the thermodynamic approach developed in this paper enables a rigorous analysis of critical dynamics in a wide class of urban systems, as well as quantitative explorations of diverse “what-if” scenarios with respect to a generic and precise efficiency measure. |
Our model is based on the Boltzmann-Lotka-Volterra (BLV) method The BLV models involve two components: a fast equilibration, “Boltzmann”, component and a slow dynamic, “Lotka-Volterra”, component. The Boltzmann component applies maximum entropy principle to derive the static flow patterns of commodities and residents c... |
The Lotka-Volterra component evolves the spatial distribution and the flow pattern of a commodity according to generalised Lotka-Volterra equations for spatially distributed competitors. In our model of Greater Sydney the Lotka-Volterra equations make suburbs compete for local services, and suburbs with more services b... |
The maximum entropy method has been applied to a variety of collective phenomena and urban modelling , suggesting a formal analogy between urban and thermodynamic systems |
In studying transformations in the Greater Sydney region as thermodynamic phenomena, we construct the corresponding phase diagram with respect to suitably chosen control parameters. In doing so we use the Fisher information, which measures the sensitivity of a probability distribution to the change in the control param... |
Our analysis further deepens the analogy between urban science and thermodynamics, utilising a clear thermodynamic interpretation of the Fisher information as the second derivative of free entropy. Specifically, we investigate the minimum work required to vary a control parameter and trace configuration entropy and int... |
Material and methods 2.1 Overview of the model In our model, the population commutes between home and work place. The number of people commuting between employment areas [MATH] and residence areas [MATH] is given by the travel-to-work matrix [MATH] Commuting trips have an associated cost [MATH] , e.g., travelling expen... |
[MATH] represents the structure of the transport network, which may include the roads as well as different types of public transport. Employment areas are characterised by the average income |
[MATH] earned by the employees that, in combination with the travel-to-work matrix, provides the flow of income [MATH] Residence areas are instead characterised by the average rent |
[MATH] , and the amount of services [MATH] (the data used in modelling Greater Sydney is described in Supplemental Material, Sec. 1). |
We develop a BLV model for the predicted income flow [MATH] in contrast with the actual income flow [MATH] obtained from the Census. The number of jobs available in each employment area is assumed to remain fixed, and therefore the income flowing out of each area is also fixed: [MATH] On the contrary, the population is... |
[MATH] of a suburb, which defines people’s preference to live in, and therefore bring their income to [MATH] When deciding where to settle, people consider the utility of living in attractive suburbs as well as the cost of commuting to work. In our model this tradeoff is controlled by two parameters, [MATH] and [MATH] ... |
The model further allows the urban services [MATH] (and therefore the attractiveness [MATH] ) to evolve, with these dynamics being slower than the resettling of people. When [MATH] units of income are moved from employment areas [MATH] to residence areas [MATH] , part of it is spent on the rent [MATH] while the remaind... |
2.2 The Boltzmann component The Boltzmann component of the model, informed by the maximum entropy principle, determines the least biased flow-of-income matrix [MATH] which satisfies the constraints on the income that employment areas can produce, the attractiveness of the residence areas and the cost of travelling. Suc... |
[EQUATION] for normalised [MATH] , subject to the constraints: [EQUATION] The constraints in ( ) fix the total income flowing out of the employment area [MATH] and towards all residence areas [MATH] The constraint in by ( ) sets the total utility [MATH] that people obtain by living in areas [MATH] with attractiveness [... |
The maximum entropy solution to this problem is [EQUATION] where [MATH] are balancing factors. The parameters [MATH] and [MATH] are the Lagrangian multipliers corresponding to the constraints in ( ) and ( ), and representing social disposition and impedance to travel, respectively. |
We calibrate our model by identifying the optimal values [MATH] and [MATH] that agree with the initial output [MATH] best matching the actual flow of income [MATH] given by Census (see Supplemental Material, Sec. 2). The evolution of the services is then modelled yielding a prediction [MATH] within Greater Sydney. |
2.3 The Lotka-Volterra component The Lotka-Volterra component of the model is given by the following dynamics for the services [MATH] over time [MATH] |
[EQUATION] where [MATH] is the total income flowing into the suburb [MATH] from all employment areas [MATH] [MATH] defines the size of the changes and [MATH] is a conversion factor such that [MATH] is the cost of running services [MATH] According to ( ), if the remaining income [MATH] (analogous to discretionary income... |
2.4 Fisher information and thermodynamic efficiency Following a recently established relationship , the rate of change of the thermodynamic work can be determined using the Fisher information (see Supplemental Material, Sec. 3): |
[EQUATION] where the Fisher information was calculated over the parameter [MATH] (fixing the parameter [MATH] ) as [EQUATION] for the maximum entropy solution [MATH] |
Finally, we define the thermodynamic efficiency of urban transformation, for a given value of [MATH] , as the reduction of entropy from the expenditure of work: |
[EQUATION] This quantity corresponds to a change [MATH] and hence relates to a transformation This approach is motivated by the notion of thermodynamic efficiency of computation |
Results 3.1 Abrupt urban transformations We explore the model predictions [MATH] over a range of values of the control parameters around their optimal values [MATH] and [MATH] We then compute the entropy [MATH] for the considered points within the phase diagram, tracing how the income distribution changes with respect ... |
However, in order to rigorously localise the abrupt change in the dynamics of income flow with respect to [MATH] , we fix [MATH] at the optimal value [MATH] and compute the Fisher information over the phase space of [MATH] The result is shown in Fig. , which shows that the Fisher information peaks at [MATH] This indica... |
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