Introduction

In physics, friction is defined as a phenomenon in which two or more objects move in the tangential direction of the contact surface or have a relative movement tendency; The force between the contact surface is friction force, which hinders the relative movement of objects on the contact surface. Prehistoric people discovered two aspects of friction: the use of friction to produce heat and the use of lubricants to reduce drag in heavy transport. Friction is widespread in the natural sciences (e.g., the assembly and operation of machinery, the formation of mountains, and the construction of houses), and also in the social sciences (e.g., political differences and cultural boycotts).

Currently, various social forms are connected and rapidly communicated globally and countries worldwide are experiencing extensive and profound social changes (Huang, 2019). Since the beginning of the 21st century, the rapid development of industrialization, urbanization, and economic globalization has led to further transformation of the social structure. This transformation process is replete with friction caused by the distribution and maintenance of benefits, cultural exchanges, collisions, and technological innovation, which have led to social instability and turbulence (Zhu, 2016). Recently, the study of social friction has increased with the emergence of new concepts and scholarly interest in practical applications and solutions for public service problems (Madsen et al. 2021). For example, Davis et al. (2019) explored how spatial and social friction influence citizens’ selection of restaurants in New York City using review data from Yelp (the biggest online review site in the United States). Luqman et al. (2020) explored how the overuse of smartphones has led to technical friction, including friction between technology and family, technology and work, and technology and personal health. Jeng and He (2022) investigated data sharing in social science research in the United States and proposed that friction during data sharing is mainly due to the time and labor intensiveness of ensuring data privacy. Nguyen and Turner (2023) analyzed the livelihood and mobility friction between APP-based drivers and traditional motorbike taxi drivers in Vietnam’s capital city Hanoi. Furthermore, based on Vietnam’s motorbike-centric urbanism, Jamme (2023) proposed the concept of productive friction, which suggests that the opportunities for street commerce depend not on the mode of transportation but on the mode of interaction between people in motion and the built environment. It can be seen that scholars have studied friction in various fields of social life (including politics, economics, and culture) and its potential effects on social development. However, due to the lack of systematic global thinking, the types and structures of social friction phenomena have not been clarified.

Physics is a discipline that studies the general laws of motion and the basic structure of matter in the real world (Hofkirchner, 2017). With the development of statistical physics, network science, data analysis, and information technology, the application of physical laws in social science has given birth to several emerging disciplines. These include economic physics and social physics, which explore the development and evolution process of social systems based on physics methods (Jusup et al. 2022). A comprehensive view of the development of the social sciences indicates that learning from and imitating natural science paradigms such as Newton’s mechanics and Darwin’s theory of evolution enables social scientists to maintain their scientific legitimacy. This approach provides a new method for systematically sorting out the types and structural characteristics of social friction phenomena from the perspectives of social physics and management mechanics. Therefore, this study aims to clarify the general definition and characteristics of social friction phenomena, construct a social friction force model based on the principles of friction force and classical mechanics, and analyze the emergence and evolution of social friction phenomena from the mechanics perspective.

Literature review

Related works on social physics

Social physics is an interdisciplinary discipline in which methods have been widely applied to the study of complex problems. Pentland (2014) defined social physics as “a quantitative social science that describes reliable mathematical connections between information, thoughts, and human behavior.” Da Luz et al. (2023) proposed that the purpose of social physics is to explore how large groups of human beings behave and provide scientific explanations for society-related processes. Specifically, social physics aims to study complex social problems through strict mathematical rules, computational reasoning, and axiomatic laws and to make rigorous interpretations of social problems in a quantitative system (Liu, 2019).

As a result of population growth, high-density agglomeration, and a high degree of organization and specialization in society, it is difficult to understand the complex and dynamic evolution of modern society solely through social science theories and methods. As a result, scholars are beginning to analyze social problems using physical science methods. Thus, social physics is an important discipline for the study of complex social problems. For example, Dima et al. (2015) studied financial kinetic energy and financial potential energy, and proposed the financial volatility model of the stock index by measuring the influence of endogenous and exogenous behaviors on system dynamics. Diep et al. (2016) constructed a social-physical model of conflicts between groups and pointed out the conflicts that could be explained by the model. Capraro and Perc (2018) clarified how statistical physics can be used to study ethical behavior. Considering the accelerated problem of uneven distribution of wealth during COVID-19, Kim et al. (2021) put forward a fair distribution method based on physics, which proposed that the linear programming problem for calculating “nucleolus” (a solution for the fair distribution) can be solved by regarding it as a physical system in which is affected by gravity.

Physical methods play an important role in improving our understanding of social challenges and finding innovative solutions. The methods of social physics also play an important role in exploring the development of human society, social governance mechanisms, and individual psychological behavior, and have accumulated rich research results. Many scholars have studied social issues from the perspective of social physics, and the combination of physics and social science has become an important part of interdisciplinary research.

Related works on social friction

As early as 1957, Reay used the term “friction” to describe the antagonism between women from different clans (Reay, 1957). Subsequently, the third revolution of science and technology greatly promoted changes in the economic, political, and cultural fields in human society. The rapid development of the Internet has been a great convenience for shopping for clothing, food, housing, and transportation, but it has also created new challenges such as social conflict, inequality, and ethical issues (Lecomte, 2023). As a result, scholars now use “social friction” to describe and analyze new challenges in the social development process. For example, Nolen-Hoeksema et al. (1997) used social friction as a concept to refer to problems, such as not feeling understood by others and family conflicts, that individuals experience in their social networks. De Soysa (2011) regarded social friction as state repression of human rights, a form of one-sided violence caused by diverse preferences. Goldfarb et al. (2015) used social friction to explain the influence of interpersonal exchange on the types of products purchased by consumers, which imposes a heterogeneous cost on purchasing some products but not others. Miloyan and Pachana (2016) defined social friction as arguments with friends, family, or colleagues.

Friction phenomena can be found in each type of social system, including the individual, institution, and environment. Social friction is always manifested as a force that works to stop or slow social development. These include individual growth, economic development, and technology innovation. For example, Fotuhi et al. (2022) distinguished the friction phenomena in the college and financial aid application process for low-income students as psychological friction and behavioral friction; the former arises from the threats to self-integrity that low-income students experience when considering attending college, and the latter refers the institutional and bureaucratic barriers that students must overcome in the process of college application. In addition, the form and content of social friction change with the development of technology and social institutions. The development of big data, cloud computing, blockchain, artificial intelligence, and other technologies has led to new kinds of social friction (e.g., technology friction, data friction, and linking friction). Edwards (2010) indicated that data friction is a form of resistance found in the processing of data travel that increases the cost of time, energy, and attention required to collect, check, store, move, receive, and access data. Hwang et al. (2022) pointed out that the social and information frictions (e.g., cultural, and linguistic differences, and limited media coverage) in virtual healthcare inhibit the free flow of remote consultations.

Related works on social mechanics modeling

Mechanics is a physics concept that is widely used in the study of social physics. In 1897, Emile Durkheim demonstrated the existence of social force, a self-inherent force as real as the force between celestial bodies, using statistical methods. Helbing and Molnár (1995) proposed a social force model describing pedestrian behavior. This model regards pedestrians as the mass point that satisfies the law’s mechanical motion; the physical concept of force describes the true force and internal motivation of pedestrians. Koster et al. (2013) pointed out a more concise social force model that solves problems related to stability and numerical resolution cost-effectively and efficiently. Wang et al. (2011) studied passenger characteristics in train stations during Spring Festival travel in China and established a simulation model of crowd movement during travel based on the social force model. Considering the direct response of individuals to information in the environment, Li and Jiang (2012) proposed a novel friction based on the social force model that focused on individual initiative. Liu et al. (2013) reviewed existing public opinion dynamics models from the perspectives of mathematics, physics, and complexity science, and suggested that future studies could model and simulate social public opinion based on social physics methods. Chen et al. (2014) summarized the research status, existing problems, and basic principles of the social force model and applied the gear prediction correction method from molecular dynamics to the social force simulation model. In doing so, the authors determined the minimum time step of the pedestrian simulation display method per Courant’s conditions.

Social physics is used to respond to complex social problems involving aspects of society such as the economy and cultural development. Social physics has accumulated a vast body of knowledge providing an important theoretical basis for promoting scientific innovation and solving complex social problems. Scholars have also adopted mechanics modeling, which includes strict mathematical laws, computational reasoning, and axiom laws, to study social issues, thereby integrating physical science methods into social science research. The successful application of these laws of physics to social science research provides a theoretical foundation for this study.

The structure analysis of social friction phenomena

Methods and materials

Grounded theory aims to help researchers develop theories using original data. Its essence is to gradually expand the abstract levels of concepts and their relationships in a spiral cycle through scientific logic, induction, comparison, and analysis, and finally forming new concepts or theories (Junek, 2012). Grounded analysis can be used to conduct a systematic qualitative analysis of the collected materials and to test theories that have been only temporarily verified; it can also be applied to the immature research field of social friction. To ensure the quality of the selected literature, the data used in this study come from the Web of Science (WOS) database and China National Knowledge Infrastructure (CNKI). WOS contains more than 8,000 high-quality journals that are the most influential in the world and are peer-reviewed. Its authority and importance have been widely recognized by the international academic community. CNKI is the world’s largest Chinese knowledge portal and the most authoritative and professional Chinese database. This study selected the Social Science Citation Index and the Science Citation Index from WOS, as well as the core journal database from CNKI. The search was conducted using “friction” as the topic word, and the literature searched was mainly from the fields of social science, philosophy, humanities, economics, management science, and so on. The search was conducted through the year 2023. The literature retrieval and sampling process are shown in Fig. 1. Finally, 106 Chinese articles and 120 English articles were selected as original data for the grounded analysis of this study.

Fig. 1: Identification process for relevant studies.
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The figure shows the selective process of materials used for qualitative analysis. The first step was the retrieval process of both Chinese and English literatures, the second process was the screening process. CNKI China National Knowledge Infrastructure, WOS Web of Science, TS topic word, SCI Science Citation Index Expanded, SSCI Social Sciences Citation Index.

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Qualitative analysis

We obtained the core sentences and high-frequency words closely related to social friction in the selected literature, and we used grounded theory for three-level coding, which is the most influential model of theory construction used by qualitative researchers in the social sciences today (Denzin, 2010). Based on this, and through in-depth interviews with ten social science experts, we clarified the structural characteristics of social friction phenomena. Specifically, the first task in the grounded analysis was to encode relevant content by carefully reading the articles obtained in the search. Three researchers were asked to code the data independently and to check each other to ensure the reliability of the coding work. Then, the literature content was conceptualized through coding analysis. Finally, through the analysis and summary of the concept, the main category was identified. On this basis, the coding results were verified and supplemented by expert interviews. In the end, 48 initial codes, 11 main categories, and 3 core codes were created. Table 1 displays the coding scheme used and the articles used in each category.

Table 1 Examples of literature selection and the coding process of social friction phenomena.
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Based on the assumptions that people are the fundamental elements of the social system, social friction can be understood as “multiple conflicts” between people, people and nature, people and institutions, people and technology. These multiple conflicts have existed in almost every aspect of life and social production, such as international relationships (interpersonal friction), technological innovation (information/data friction), economic development (market/trade friction), and social transformation (institutional friction). These phenomena do not necessarily become a form of resistance to social development. Friction is often necessary for human development and meaningful engagement with others (Frischmann, 2022). A certain degree of social friction can improve the quality of decision-making and promote social transformation and economic development. Wei et al. (2019) found that Sino–U.S. trade friction is conducive to optimizing the employment structure of Chinese companies and accelerating high-quality innovation. Liu et al. (2020) showed that trade frictions can reduce the greenhouse gas emissions of participants and improve air quality in most countries. However, excessive social friction will hinder the process of social development, generate friction costs (including economic losses and non-economic losses), and even trigger mass incidents. Higher financial friction reduces the overall efficiency of investment and total output by hindering the reallocation of resources among heterogeneous entrepreneurs (Wang, 2023). Alternatively, high social friction resulting from diverse preferences and coordination failures can also contribute to poverty (De Soysa, 2009).

To further analyze the characteristics of social friction, this study distinguished and defined the two-dimensional structure of “explicit–implicit” social friction phenomena, as shown in Fig. 2. Implicit friction is the internal root of social friction and includes interpersonal friction, cognitive friction, cultural friction, information friction, and technological friction. Interpersonal friction occurs when an individual feels stressed, experiences affective misalignment, is misunderstood or unsupported, or has an argument with friends, family, and colleagues (Miloyan and Pachana, 2016; Shimizu and Nakashima, 2017; Saigot, 2022). Cognitive friction involves human psychology and emotions and refers to the ideological differences caused by demographic characteristics, knowledge gaps, and cognitive deviation (Chen, 2017; Qiu et al. 2020). Cultural friction refers to cultural resistance caused by conflicts in value systems, religious beliefs, traditional customs, and behavior patterns (Luo and Shenkar, 2011). Cognition is the most basic psychological process in human beings and involves information processing of perception, memory, imagination, and thinking and has an important effect on individuals’ behavior choices. Culture is a complex entity composed of values, social rules, lifestyles, patterns of interpretation and behavior (Schachtner, 2015), and abilities and habits acquired as a member of society. Culture is the basis for human survival and development. Cognitive friction and cultural friction are the core components of the social friction phenomena and have an inseparable intrinsic relationship to other types of social friction. Information friction occurs in economics and refers to the bid-ask spread caused by information asymmetry or uncertainty between the parties to a transaction (Thierry et al. 2007), usually increasing the cost of time and labor. Technological friction refers to differences and disputes between countries on scientific and technological issues that may lead to political and economic antagonisms and negative outcomes in international relations.

Fig. 2: The structure of social friction phenomena and their interactive relationships.
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The figure shows the two-dimensional structure of “explicit and implicit” social friction phenomena and the interactive and evolutionary relationship between implicit social friction and explicit social friction.

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Explicit friction is the external manifestation of social friction and includes behavioral friction, migration friction, economic friction, and institutional friction. Behavioral friction refers to contradictions and conflicts caused by ideological concepts, lifestyles, etc. For example, in the prevention and control of COVID-19 in 2020, many “small frictions” occurred between residents and community workers. As residents had limited knowledge and awareness of preventing and controlling the spread of the virus, many did not actively cooperate with cumbersome prevention and control measures and pushed, insulted, and slandered community workers (Yu and Bai, 2020). Migration friction refers to the cost of settlement restrictions caused by relocation and the household registration system. It also includes psychological and behavioral obstacles, such as changes in behavior and a sense of not belonging caused by the relocation and household registration system (Liu and Li, 2017). Migration friction hinders the free flow of populations, distorts the balanced distribution of urban population space, and causes utility losses (Behrens et al. 2016). Economic friction is a much-studied topic in the social sciences, especially since the Sino-US trade war in 2018. Economic friction refers to conflicts of interest between regions/countries due to political differences or ideological conflicts. These differences can lead to economic losses and lower levels of social welfare, and also financial and trade frictions. Institutional friction refers to the structural barriers between regions/countries caused by institutional differences/ambiguities, political changes, ideologies, and other factors, including horizontal institutional frictions and vertical institutional frictions. In addition, the essence of international economic friction is also a kind of institutional friction and manifests as friction within the institution. It is also the interest game relationship between economic subjects under the new institution (He and Lu, 2015).

In general, there are interactive and evolutionary relationships between implicit and explicit social frictions. Implicit friction is the root cause of explicit friction, and explicit friction is the external manifestation of the accumulation of implicit friction. Both implicit friction and explicit friction generate social friction costs.

The construction of the social friction force model

Based on the theories and methods of social physics, this section includes the following: presents content on friction force, classical mechanics, physics-related theoretical methods, and effective expansion and rational correction; constructs a model of social friction force; conducts a mechanical analysis of social friction; and discusses its contribution to social development and evolution.

The theories of friction force and classical mechanics

The essence of friction force is the vector sum of many factors acting on atoms on the surface of more than one object. Generally speaking, friction force includes static friction force and dynamic friction force. Static frictional force is the resistance of an object to another object when it has a relative motion tendency on the surface of another object. Dynamic friction force refers to a force generated on the contact surface between two objects that hinders relative movement. Dynamic friction force can be a driving force or an obstructing force. Benefits friction can generate a desired reaction. For example, the friction between shoes and the ground that propels people to walk or run and the friction between tires and the road that propels a car forward or backward. Harmful friction can cause unfavorable wear on objects causing energy loss. For example, the friction between chalk and a blackboard and the wearing down of the chalk. Too much friction will generate heat and resistance, but too little friction will have the opposite result. For example, when car wheels slip due to insufficient friction between car tires and the ground (Luo and Shenkar, 2011).

In physics, force, work, and energy are three closely related concepts. Force is the root cause of the change or deformation of an object’s motion state, work is a physical quantity that represents the accumulation of space in which a force acts on an object, and energy is a measure of the ability of a physical system to do the work. Work is the process of transforming energy from one form to another; how much work is done must be accompanied by the transfer or transformation of energy. According to the definition of classical mechanics, when a force acts on an object and causes the object to pass a certain distance in the direction of the force, it is said that the force does work on the object. Work is a scalar; there is no direction, but there are positive and negative points. As shown in Fig. 3, when \(W > 0\), the friction force \({f}\) does positive work and can be regarded as a kind of driving force. When \(W < 0,\) the friction force \(f\) does negative work and can be regarded as a kind of obstruction force. When \(W=0\), the friction force \(f\) does no work.

Fig. 3: Schematic diagram of friction force and related principles of classical mechanics.
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The figure shows the theories of friction force and work in physics. The first two parts were the basic principles of friction force in physics, the third part was the work done by friction force. M represents the object, G represents gravity, F represents normal force, f represents friction force, W represents the work done by friction force, \(\mu\) represents the dynamic friction factor.

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Comparability of physics friction and social friction

In this section, we discuss the similarity and comparability of friction in physics and friction in social systems from three aspects. These are the preconditions of how the theories and methods of friction force and classical mechanics can be used to develop the social friction force model.

First, the comparability of the physical system and social system is discussed. In physics, a system is a whole composed of many particles as per certain rules. The basic components of a social system are not particles but humans as human society is composed of many individuals with a consciousness who interact. The interconnections among individuals constitute society (Wang, 2020). It is worth noting that the social system, composed of people, and the physical system, composed of particles, have a similar framework. The behavior of individuals in the social system is similar to the behavior of particles in a physical system. For example, Bouvier et al. (1997) constructed a crowd movement model based on a particle system, that considered the whole crowd as a particle system, in which each person was a particle that interacted with other particles. The movement of individuals in the social system is similar to the movement of particles in the physical system. The movement of particles in the physical system follows the laws of dynamics. Specifically, classical particles follow the laws of Newton’s mechanics (Kadupitiya et al. 2022); microscopic particles follow the laws of quantum mechanics (Han et al. 2014). However, both classical and microscopic particles derive from the variation principle, in which, the motion of particles is determined by taking an extreme value of the corresponding universal function. For example, in a social system, buyers and sellers in a market also pursue the highest exchange value of products based on a generalized variation principle. Moreover, particles must interact with each other (e.g., collisions) to bring the system into equilibrium and obtain macroscopic properties, such as a uniform temperature. Similarly, there is a correlation of information transmission between exchangers in the market that is equivalent to particle collisions. The exchange value of different buyers and sellers evens out, producing a unified market price. De Domenico et al. (2015) regarded individuals as the particles in a thermodynamical system and modeled human mobility behavior in an urban system as interacting particles with data-driven origin-destination pairs whose trips can be routed. The social system and physical system have a certain degree of homogeneity in the internal correlation of system evolution (Yin and Wang, 2016). This comparability between physical and social systems is not a coincidence but reflects the universality of large interacting systems. In addition, the similarity between the elements of physical and social systems is the basis of the comparability method.

Second, the application of physical models and techniques (e.g., the law of universal gravitation, kinetic theory, combustion, and entropy) to social science research has produced many classical theories, such as social combustion theory, social behavior entropy theory, social shock wave theory, and social force model (Liu et al. 2009; Korecki et al. 2016). These theories have been widely used in the study of social problems. For example, Wu et al. (2022) modeled crowd evacuation in public places using a behavioral heterogeneity-based social force model, which has important guiding significance for formulating reasonable evacuation schemes in specific scenarios. Shan and Gao (2010) proposed a framework for the interpretation of mass incidents using the method of social combustion theory. The studies of social physics theories and their applications provide theoretical and methodological references for this study.

Third, the feasibility of analyzing social friction using the friction force model is discussed. In physics, the basic condition for friction force is that when two objects are in direct contact and compress together, the contact surface is rough, and there is relative movement or relative movement tendency. In a social system, the various specific activities that people are engaged in are inseparable from the social forms, relationships, and institutions formed between people. The social system is composed of people, finances, materials, energy, information, and so on. With the rapid development of the social economy, contact between people is growing more frequent. The widespread connection between people and people and the environment in the social system is similar to the basic condition for friction force in physics (two objects are in direct contact and are compressed together). In addition, dynamics, disorder, change, and diversity are the core features that promote the operation of social systems. However, the above features can also push the social system to break out or surpass the original equilibrium state thus storing energy and creating an innovative dynamic structure in the social system. Exclusivity is an instinct of people in the group, and society is not always harmonious and stable. However, the various conflicts and disorders in human society are similar to the basic conditions for friction force in physics (the contact surface is rough).

Finally, the social competition between individuals or groups for higher social status or more benefits has become an important driving force for social development. From this point of view, there must be relative movement, or relative movement tendency, among the various elements in the social system.

Modeling of social friction force

As discussed above, the social system is similar to the physical system, and many physics theories and methods have been used to explore social problems. Furthermore, the various components of the social system meet the basic conditions of contact and compression, rough contact surface, and relative movement or relative movement tendency. Therefore, the contradictions and conflicts generated in social activities can be studied by constructing a social friction force model. Social friction is a state of contradiction or conflict between the various components of the social system due to competition during the process of social production and living. Social friction force measures the intensity of these contradictions and conflicts, which can be calculated using the following equations:

$${f}_{S}=\alpha \,{{\cdot }}\,{F}_{{ij}}+\epsilon$$
$${F}_{{ij}}=\beta {F}_{{PS}}+\gamma {F}_{{PH}}$$

Where α is the social friction coefficient, which is determined by the intrinsic properties of the social system in a specific period. Based on the social force model modified by Wang et al. (2011), \({F}_{{ij}}\) is the force between two interacting elements in the social system, \(i\) and \(j\); \(\epsilon\) is the random error term; \(\beta\) and \(\gamma\) are constants, which are set in different models according to their specific situation; \({F}_{{PS}}\) represents mental force; and \({F}_{{PH}}\) signifies physical force.

Compared with the relevant principles of work done by friction force in physics, the work done by social friction force was defined as a “contribution” to society (represented by \(W\) in the equation below). Different from physics, the work done by social friction force does not cause a significant “displacement” in the social system. However, a certain period of time must pass between the generation and transfer of the social friction force. Additionally, the magnitude of the social friction force must be related to time. Social friction force is a dynamic force that evolves with social development level. Therefore, compared with the micro-element calculation method of variable force work, the magnitude of social friction force can be expressed as follows:

$$W={\int}_{\!0}^{t}{f}_{\!s}{dt}$$

Where the work done by social friction force must not be 0 (\(W\ne 0\)). Specifically, if \(W > 0,\) the social friction force does positive work, which can also be seen as a “positive contribution” to the development of a social system. If \(W < 0\), the social friction force does negative work, which can also be seen as a “negative contribution” to the development of a social system. Magnitude and direction are two important factors that determine the work done by the friction force. As shown in Fig. 4, \({f}_{\max }\) was assumed to be the maximum static friction force that an orderly organization can bear under the steady-state society. The specific conditions of work done by social friction force are as follows:

Fig. 4: Mechanical analysis of social friction phenomena.
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The figure shows the evolution process of social friction force and social friction phenomena over time. \({F}_{N}\) represents the normal force, \(\mu\) represents the dynamic friction factor, f represents friction force; \({F}_{{ij}}\) represents the force between two interacting elements (i and j) in the social system, α represents the social friction coefficient, ϵ represents the random error term, \({f}_{s}\) represents social friction force, \({f}_{\max }\) represents the maximum static friction force, t represents time; W represents the work done by social friction force \({f}_{s}\), which was defined as a kind of “contribution” to society in this study.

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Where during the period \([0,{t}_{1}]\), social friction force \({f}_{S}\) is less than the maximum social static friction force \({f}_{\max }\), and social organization is in a stable and orderly development state (which can also be defined as homeostasis). During this period, the social friction force \({f}_{S}\) manifests as a static friction force, and the corresponding social friction phenomena are mainly manifested as implicit social friction (e.g., cultural friction, cognitive friction, information friction, and technological friction). When \({f}_{S}\) is less than \({f}_{\max }\), the implicit friction has a positive effect on the development of social organizations. For example, cultural friction posits that cultural differences may be either synergistic or disruptive. For example, synergies will emerge in foreign-invested enterprises when cultural differences in leadership beliefs are less central to the host nation’s cultural identity (Koch et al. 2016). Unlike the distance metaphor, friction captures actual inter-partner conflicts, allowing for the possibility of finding both underlying reasons and effective remedies for conflict reduction and attachment enhancement. In addition, cognitive friction and technological friction, represented by different ideological concepts and scientific and technological disputes, are important sources of power for the generation of new ideas and technological innovation. Therefore, when social friction force is manifested as a static friction force, the existence of implicit social friction will help promote the development of social organizations. Specifically, it will be expressed as a “positive force” that is conducive to promoting technological innovation and social governance system reforms. Its direction is the same as the positive development direction of socially orderly organizations. This means that the social friction force does “positive work” (\({W}_{1} > 0\)) and makes a “positive contribution” to the development of social organizations. Consequently, during this period, the various subjects of social friction should adopt active cooperation strategies to give full play to the synergy of social friction phenomena, as well as its “positive contribution” to the development and innovation of organizations. Meanwhile, various subjects should also adopt corresponding defense strategies to ensure that the level of social friction force is less than the critical threshold that the maximum social static friction force \({f}_{\max }\) can withstand.

Second, with the accumulation of social inequities and social contradictions, social friction force \({f}_{S}\) exceeds the critical threshold \({f}_{\max }\) that an orderly organization of society can support, with the result that the existing operating environment of the subjects of social friction breaks. The social friction force turns into a kind of dynamic friction force, and the corresponding social friction phenomena are manifested as explicit social friction (e.g., institutional friction, economic friction, behavioral friction, and migration friction). For example, Bao et al. (2020) found that additional tariffs imposed by the United States had a minor impact on China’s macro economy, but a significant impact on the export of high-end manufacturing industries. Bouët and Laborde (2018) pointed out that trade frictions have not only failed to improve public welfare or GDP in the United States, but have caused serious damage to trading partners. Grechyna (2016) suggested that political friction can be understood as political polarization or as public rent-seeking, which leads to public spending. In addition, migration friction is an important factor affecting the scale of cities in China and the United States (Desmet and Rossi-hansberg, 2013). Specifically, migration friction hinders the population’s free flow, distorts the population’s balance distribution, and reduces productivity or product elasticity (Behrens et al. 2016). Explicit social friction always brings friction costs (both economic and non-economic). During the period of [\({t}_{1},{t}_{3}]\), \({f}_{S}\) was greater than \({f}_{\max }\), but the direction of social friction force has different evolutionary trends, and the work done by social friction force also shows different evolutionary characteristics.

Specifically, when the direction of the social friction force is opposite the orderly development direction of social organizations (corresponding to the curves AB and BD in Fig. 4), the interest conflict between the subjects of social friction begins to intensify, and social conflict behaviors occur. When the subjects of social friction choose the negative-sum game (or zero-sum game) strategy, the dominant social friction phenomena further evolve into a group event or a public crisis event (such as the “June 28” incident in Weng’an County, Guizhou Province, in 2008, which can be regarded as a typical mass event caused by the social friction phenomena). During this period, social friction force is manifested as a kind of “negative force” that hinders social development and produces a “negative contribution” to the development of organizations (\({W}_{2}\, < \,0\)). Therefore, during this period, all subjects impacted by social friction should take effective measures to reduce social conflict behavior so as to reduce social friction force. In addition, positive policy interventions should be adopted to guide organizations to choose healthy competition and cooperative game strategies to prevent group incidents.

When the direction of social friction is the same as the development direction of the socially orderly organization (corresponding to the curve BC in Fig. 4), the social friction force at this period is greater than the maximum static friction the socially orderly organization can withstand. Therefore, the various subjects of social friction reach a consensus conducive to promoting social development by adopting positive-sum game strategy behaviors. Finally, a synergistic effect of \(1+1 > 2\) is produced. It is expressed as a “positive force” that promotes social development, and produces a “positive contribution” to social organizations development (\({W}_{3} > \,0\)). Therefore, during this period, the government should establish and improve corresponding market governance and supervision mechanisms, give full play to the positive role of social friction, and increase its positive contribution to organizational development. Meanwhile, \({f}_{\max }\) should be increased to improve the overall stability of society.

Conclusion and research prospects

This study first clarifies the “explicit and implicit” characteristics of social friction phenomena. Specifically, implicit friction is the internal root of social friction, which includes interpersonal friction, cultural friction, cognitive friction, information friction, and technical friction; explicit friction is the external manifestation of social friction and includes behavior friction, migration friction, economic friction, and institutional friction. Second, the social friction force model was extended and constructed, based on the social force model, and the general expression and expression form of social friction force were defined. When the social friction force \({f}_{s}\) is less than the maximum static friction force that the social system can bear \({f}_{\max }\), the social friction force remains a static friction force. When \({f}_{s}\) exceeds \({f}_{\max }\), the social friction force is a dynamic friction force. Third, the cumulative effect of social friction force over its acting time is defined as the “social contribution” of social friction force by the infinitesimal method based on variable force work. Specifically, when the direction of social friction force is the same as the direction of orderly social development, social friction force acts as a positive force to promote social development. When social friction force is opposite to the direction of orderly social development, social friction force acts as a “reverse force” that hinders social development.

This study developed the general concept of social friction and distinguished the explicit and implicit characteristics of social friction phenomena. Through the construction of the social friction force model, the emergence and evolution of social friction phenomena were explained using mechanical principles. The exploration of the social friction force model makes it possible to model and quantify social friction based on theories and methods used in physics. This approach provides a new theoretical perspective for the study of friction phenomena in social systems, and enriches research on social physics. This study has some limitations that future research should address. First, although the social friction force model provides new theoretical and methodological references for analyzing the friction phenomena in social systems, a limited number of parameters and rules were considered in this study. Future researchers should test and modify the social friction force model by introducing new mechanical models or reconstructing the parameters and variables of the present model. Second, this study developed a social friction force model by borrowing theories and methods in physics without analyzing the social friction phenomena quantitatively or using real data. Therefore, the measurement and calculation of social friction force are an important direction for future research. Third, this study only considered friction cost, rather than its benefits. In fact, social friction phenomena are not necessarily barriers to social development. On the contrary, friction is often necessary for human development and meaningful engagement with each other and our techno-social environments. Therefore, future research needs to further explore the impact of social friction on social development, how to prevent or manage harmful friction and how to maximize the application of beneficial friction. Finally, this study is an introduction to social friction phenomena and the social friction force model and aims to stimulate researchers’ attention and in-depth discussion on the friction phenomena in social systems.