**2. Entropy in Cryptocurrency Markets**

Thermodynamics-related theories in physics have driven the development of economics in the past 100 years, especially when the theory of 'entropy' in thermodynamics emerged, which greatly promoted the development of modern economic theories. For example, entropic economics, complex economics, and quantum economics, etc. This shows that 'entropy' has bridged the gap between economics and physics and has had a dramatic impact on mainstream economic theory. The concept of 'entropy' originated in the 19th century, and first indicated that part of the energy of a steam engine could not be transformed into useful work due to friction and other reasons, and 'entropy' measured the missing energy in this part. The first mathematical definition of 'entropy' is shown below [1].

$$\mathcal{C} = \frac{1}{T}q\tag{1}$$

$$
\Delta \mathcal{E} = \left(\frac{1}{T\_2} - \frac{1}{T\_1}\right) q \tag{2}
$$

where Δ*c* represents the change in entropy and *q* is the heat transferred from an object with temperature *T*<sup>1</sup> to other object with temperature *T*2. A slightly different definition of entropy, being a measure of the molecular disorder of the system, was formulated by Boltzmann. It has the following form:

$$
\mathbb{C} = \mathbb{K} \text{ } \ln(m) \tag{3}
$$

where *K* is the Boltzmann constant, while *m* is the number of microscopic states.

'Entropy' in thermodynamics can be used in the social sciences as a general measure of the disorder in a system. For example, the concept of 'corporate entropy' is used in management and organizational sciences, and should be understood as a loss of productive energy. The entropy in an organization is always growing, just like the thermodynamic entropy in the universe. As the concept of 'entropy' continues to evolve, the various definitions of entropy can be made more specific and applied to specific financial scenarios. John Bryant in his book provides a careful mathematical description of 'entropy' in the economy with the following expression shown in, it has the following form:

$$
\varepsilon = \ln\left(\frac{v}{L}\right) \tag{4}
$$

where *v* represents the volume of economic activity and *L* represents the constrained level of that activity. The change in the entropy of the economy per unit of time can be expressed in a more precise formula.

$$d\mathbb{C} = (w - wn + 1)\frac{dv}{v} = \left(1 + \frac{1 - n}{r}\right)\frac{dv}{v} \tag{5}$$

where *dv/v* represents the growth rate of volume flow, *w* is the lifetime factor, n represents the elasticity index, and r is the natural rate of return. The factor (*w* − *wn + 1*) is called the marginal entropy index, and the integration of Equation (5) yields the entropy generation per unit time using the following mathematical form:

$$\mathcal{L} = (w - wn + 1)\ln(v) + \mathcal{c}\_0 = \left(1 + \frac{1 - n}{r}\right)\ln(v) + \mathcal{c}\_0 \tag{6}$$

Equation (6) can be used to describe the monetary entropy, in which case the rate of return approximates the long-run average or natural level of the velocity of money circulation.

Kolmogorov entropy is an important quantity to characterize chaotic systems. In different types of dynamical systems, the value of *K* is different, and in systems with chaotic motion, the value of *K* is greater than zero. the larger the value of *K*, then the greater the rate of loss of information. the formula for Kolmogorov entropy is shown below [2].

$$k = -\lim\_{\tau \to 0} \lim\_{\varepsilon \to 0} \lim\_{d \to \infty} \frac{1}{d\tau} \sum\_{i\_1, \dots, i\_d} p(i\_1, \dots, i\_d) \ln p(i\_1, \dots, i\_d) \tag{7}$$

where *p*(*i1,* ... *, id*) is the joint probability and ε and d are fixed values. Equation (7) can portray the degree of chaos of cryptocurrency and smart cities. Samet Gunay andand Kerem [3] showed that cryptocurrency markets are not random but chaotic. In the present, this means that the short-term prediction of the cryptocurrency market may be achievable, but it is completely unpredictable in the long-term prediction.

R. Fistola and R. A. La Rocca [4] studied the measurement of urban entropy, which is a complex system, and divided the urban system into five subsystems, each of which contains several influences that are used to measure the entropy of the city. In addition, it is beneficial to keep the entropy of a city within a reasonable range, but when the entropy is too low or too high, this can lead to a 'fragile' city or reduce the ability to sustain development. Dehouche [5] uses an approximate entropy approach to verify the reasons for the exponential and persistent fluctuations of the bitcoin price, using data such as the daily price of bitcoin, the price of gold, and the SandP 500 index, and calculating their standard deviations. Pele, DT and Marinescu-Pele [6] used the entropy of bitcoin's daily returns to predict the daily value-at-risk of bitcoin and demonstrated that entropy outperforms the classical GARCH model, and the following conclusions are drawn: There is a strong positive correlation between the daily log price of bitcoin and the intra-day return entropy, indicating that entropy has predictive power for bitcoin price. Grilli and Domenico [7] introduced the concept of Boltzmann entropy into cryptographic digital currencies and used Boltzmann entropy to predict the price change trend of cryptographic digital currencies.

The entropy of the cryptocurrency market and the traditional currency market will be affected by various factors, as shown in Table 1 such as inflation rate, fiscal deficit level, interest rate volatility, and high cost of currency management in the traditional currency market. For the cryptocurrency market, the basic blockchain peer-to-peer and decentralized technology of cryptocurrency saves a lot of operation and management costs, meanwhile, along with the emergence of NFT, Dao, Web 3.0, metaverse in the cryptocurrency market, they are constantly optimizing the ecological environment of the crypto market.


**Table 1.** The factors of influencing the entropy values.

In economic reform, 'entropy' can be used as a tool for future monetary reform, in this paper mentioned cryptocurrency and smart city, in which includes a large number of 'entropy', such as: 'monetary entropy ', 'education entropy', 'transportation entropy', 'ecological entropy', etc., when cryptocurrencies appear in the monetary market as well as transforming the urban governance model to 'smart city' mode is transformed, it is in reducing the degree of chaos within the original system, reducing the lost part of the system operation, improving the operation efficiency, and then introducing 'entropy' into the currency market and city construction for the global crypto market and city governance model.

In this paper, we introduce the concept of Kolmogorov entropy to smart cities and cryptocurrency, and use Kolmogorov entropy to measure the degree of disorder in the monetary market of smart cities as a way to speculate whether the smart cities are developing stably.
