The approach towards equilibrium in a reversible Ising dynamics model -- an information-theoretic analysis based on an exact solution

We study the approach towards equilibrium in a dynamic Ising model, the Q2R cellular automaton, with microscopic reversibility and conserved energy for an infinite one-dimensional system. Starting from a low-entropy state with positive magnetisation, we investigate how the system approaches equilibrium characteristics given by statistical mechanics. We show that the magnetisation converges to zero exponentially. The reversibility of the dynamics implies that the entropy density of the microstates is conserved in the time evolution. Still, it appears as if equilibrium, with a higher entropy density is approached. In order to understand this process, we solve the dynamics by formally proving how the information-theoretic characteristics of the microstates develop over time. With this approach we can show that an estimate of the entropy density based on finite length statistics within microstates converges to the equilibrium entropy density. The process behind this apparent entropy increase is a dissipation of correlation information over increasing distances. It is shown that the average information-theoretic correlation length increases linearly in time, being equivalent to a corresponding increase in excess entropy.Comment: 15 pages, 2 figure

Expressing the entropy of lattice systems as sums of conditional entropies

Whether a system is to be considered complex or not depends on how one searches for correlations. We propose a general scheme for calculation of entropies in lattice systems that has high flexibility in how correlations are successively taken into account. Compared to the traditional approach for estimating the entropy density, in which successive approximations builds on step-wise extensions of blocks of symbols, we show that one can take larger steps when collecting the statistics necessary to calculate the entropy density of the system. In one dimension this means that, instead of a single sweep over the system in which states are read sequentially, one take several sweeps with larger steps so that eventually the whole lattice is covered. This means that the information in correlations is captured in a different way, and in some situations this will lead to a considerably much faster convergence of the entropy density estimate as a function of the size of the configurations used in the estimate. The formalism is exemplified with both an example of a free energy minimisation scheme for the two-dimensional Ising model, and an example of increasingly complex spatial correlations generated by the time evolution of elementary cellular automaton rule 60

War of attrition with implicit time cost

In the game-theoretic model war of attrition, players are subject to an explicit cost proportional to the duration of contests. We construct a model where the time cost is not explicitly given, but instead depends implicitly on the strategies of the whole population. We identify and analyse the underlying mechanisms responsible for the implicit time cost. Each player participates in a series of games, where those prepared to wait longer win with higher certainty but play less frequently. The model is characterised by the ratio of the winner's score to the loser's score, in a single game. The fitness of a player is determined by the accumulated score from the games played during a generation. We derive the stationary distribution of strategies under the replicator dynamics. When the score ratio is high, we find that the stationary distribution is unstable, with respect to both evolutionary and dynamical stability, and the dynamics converge to a limit cycle. When the ratio is low, the dynamics converge to the stationary distribution. For an intermediate interval of the ratio, the distribution is dynamically but not evolutionarily stable. Finally, the implications of our results for previous models based on the war of attrition are discussed.Comment: Accepted for publication in Journal of Theoretical Biolog

Induced Technological Change in a Limited Foresight Optimization Model

The threat of global warming calls for a major transformation of the energy system the coming century. Modeling technological change is an important factor in energy systems modeling. Technological change may be treated as induced by climate policy or as exogenous. We investigate the importance of induced technological change (ITC) in GET-LFL, an iterative optimization model with limited foresight that includes learning-by-doing. Scenarios for stabilization of atmospheric CO2 concentrations at 400, 450, 500 and 550 ppm are studied. We find that the introduction of ITC reduces the total net present value of the abatement cost over this century by 3-9% compared to a case where technological learning is exogenous. Technology specific polices which force the introduction of fuel cell cars and solar PV in combination with ITC reduce the costs further by 4-7% and lead to significantly different technological solutions in different sectors, primarily in the transport sector.Energy system model, Limited foresight, Climate policy, Endougenous learning, Technological lock-in

Bifurcation in Quantum Measurement

We present a generic model of (non-destructive) quantum measurement. Being formulated within reversible quantum mechanics, the model illustrates a mechanism of a measurement process --- a transition of the measured system to an eigenstate of the measured observable. The model consists of a two-level system $\mu$ interacting with a larger system $A$, consisting of smaller subsystems. The interaction is modelled as a scattering process. Restricting the states of $A$ to product states leads to a bifurcation process: In the limit of a large system $A$, the initial states of $A$ that are efficient in leading to a final state are divided into two separated subsets. For each of these subsets, $\mu$ ends up in one of the eigenstates of the measured observable. The probabilities obtained in this branching confirm the Born rule.Comment: A revised version that includes a more general presentation of the model (in Sect. 4) and a larger revision of the Introductio

Financing the Transition Toward Carbon Neutrality—an Agent-Based Approach to Modeling Investment Decisions in the Electricity System

Transitioning to a low-carbon electricity system requires investments on a very large scale. These investments require access to capital, but that access can be challenging to obtain. Most energy system models do not (explicitly) model investment financing and thereby fail to take this challenge into account. In this study, we develop an agent-based model, where we explicitly include power sector investment financing. We find that different levels of financing constraints and capital availabilities noticeably impact companies\u27 investment choices and economic performances and that this, in turn, impacts the development of the electricity capacity mix and the pace at which CO2 emissions are reduced. Limited access to capital can delay investments in low-carbon technologies. However, if the financing constraint is too relaxed, the risk of going bankrupt can increase. In general, companies that anticipate carbon prices too high above or too far below the actual development, along with those that use a low hurdle rate, are the ones that are more likely to go bankrupt. Emissions are cut more rapidly when the carbon tax grows faster, but there is overall a greater tendency for agents to go bankrupt when the tax grows faster. Our energy transition model may be particularly useful in the context of the least financially developed markets

Modelling the Transition towards a Carbon-Neutral Electricity System—Investment Decisions and Heterogeneity

To achieve the climate goals of the Paris Agreement, greenhouse gas emissions from the electricity sector must be substantially reduced. We develop an agent-based model of the electricity system with heterogeneous agents who invest in power generating capacity under uncertainty. The heterogeneity is characterised by the hurdle rates the agents employ (to manage risk) and by their expectations of the future carbon prices. We analyse the impact of the heterogeneity on the transition to a low carbon electricity system. Results show that under an increasing CO2\ua0tax scenario, the agents start investing heavily in wind, followed by nuclear and to some extent in natural gas fired power plants both with and without carbon capture and storage as well as biogas fired power plants. However, the degree to which different technologies are used depend strongly on the carbon tax expectations and the hurdle rate employed by the agents. Comparing to the case with homogeneous agents, the introduction of heterogeneity among the agents leads to a faster CO2\ua0reduction. We also estimate the so called “cannibalisation effect” for wind and find that the absolute value of wind does not drop in response to higher deployment levels, but the relative value does decline